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If the length of the latus rectum of the ellipse \( x^{2}+4y^{2}+2x+8y-\lambda=0 \) is \( 4 \), and \( l \) is the length of its major axis, then \( \lambda + l \) is equal to _______.

An angle of intersection of the curves, \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) and \( x^{2} + y^{2} = ab \), \( a > b \), is:

Consider ellipses \( E_k : kx^2 + k^2y^2 = 1, \, k = 1, 2, \ldots, 20 \). Let \( C_k \) be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse \( E_k \). If \( r_k \) is the radius of the circle \( C_k \), then the value of \( \displaystyle \sum_{k=1}^{20} \frac{1}{r_k^2} \) is:

The centre of a circle \( C \) is at the centre of the ellipse \( E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \). Let \( C \) pass through the foci \( F_1 \) and \( F_2 \) of \( E \) such that the circle \( C \) and the ellipse \( E \) intersect at four points. Let \( P \) be one of these four points. If the area of the triangle \( PF_1F_2 \) is \( 30 \) and the length of the major axis of \( E \) is \( 17 \), then the distance between the foci of \( E \) is:

If the point on the curve \( y^{2}=6x \) nearest to the point \( \left(3,\dfrac{3}{2}\right) \) is \( (\alpha,\beta) \), then \( 2(\alpha+\beta) \) is equal to _____.

Let the sum of the focal distance of the point \( P(4,3) \) on the hyperbola \( H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1 \) be \( 8\sqrt{\dfrac{5}{3}} \). If for \( H \), the length of the latus rectum is \( l \) and the product of the focal distances of the point \( P \) is \( m \), then \( 9l^2+6m \) is equal to:

Let \( y=p(x) \) be the parabola passing through the points \( (-1,0), (0,1) \) and \( (1,0) \). If the area of the region \( \{(x,y):(x+1)^2+(y-1)^2\le 1,\; y\le p(x)\} \) is \( A \), then \( 12(\pi-4A) \) is equal to ________.

If the ellipse \( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \) meets the line \( \dfrac{x}{7}+\dfrac{y}{2\sqrt{6}}=1 \) on the x-axis and the line \( \dfrac{x}{7}-\dfrac{y}{2\sqrt{6}}=1 \) on the y-axis, then the eccentricity of the ellipse is:

Let \( E_1 : \frac{x^2}{9} + \frac{y^2}{4} = 1 \) be an ellipse. Ellipses \( E_i \)'s are constructed such that their centres and eccentricities are same as that of \( E_1 \), and the length of minor axis of \( E_i \) is the length of major axis of \( E_{i+1} \) \( (i \geq 1) \). If \( A_i \) is the area of the ellipse \( E_i \), then \( \frac{5}{\pi}\left( \displaystyle \sum_{i=1}^{\infty} A_i \right) \) is equal to ______.

Let the focal chord of the parabola \(P: y^{2}=4x\) along the line \(L: y=mx+c,\, m>0\) meet the parabola at the points \(M\) and \(N\). Let the line \(L\) be a tangent to the hyperbola \(H: x^{2}-y^{2}=4\). If \(O\) is the vertex of \(P\) and \(F\) is the focus of \(H\) on the positive \(x\)-axis, then the area of the quadrilateral \(OMFN\) is:

The parabola \( y^2 = 4x \) divides the area of the circle \( x^2 + y^2 = 5 \) in two parts. The area of the smaller part is equal to:

On the ellipse \( \frac{x^2}{8} + \frac{y^2}{4} = 1 \) let \( P \) be a point in the second quadrant such that the tangent at \( P \) to the ellipse is perpendicular to the line \( x + 2y = 0 \). Let \( S \) and \( S' \) be the foci of the ellipse and \( e \) be its eccentricity. If \( A \) is the area of the triangle \( SPS' \) then, the value of \( (5 - e^2)A \) is:

Let \( PQ \) be a focal chord of the parabola \( y^2 = 4x \) such that it subtends an angle of \( \pi/2 \) at the point \( (3, 0) \). Let the line segment \( PQ \) be also a focal chord of the ellipse \( E : \, \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \, a^2 > b^2 \). If \( e \) is the eccentricity of the ellipse \( E \), then the value of \( \frac{1}{e^2} \) is equal to:

Let \( P \) be a parabola with vertex \( (2,3) \) and directrix \( 2x+y=6 \). Let an ellipse \( E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1, a>b \), of eccentricity \( \dfrac{1}{\sqrt{2}} \) pass through the focus of the parabola \( P \). Then, the square of the length of the latus rectum of \( E \), is;

Let a line \( L \) pass through the point of intersection of the lines \( bx+10y-8=0 \) and \( 2x-3y=0, \, b\in \mathbb{R}-\left\{ \dfrac{4}{3} \right\} \). If the line \( L \) also passes through the point \( (1,1) \) and touches the circle \( 17(x^{2}+y^{2})=16 \), then the eccentricity of the ellipse \( \dfrac{x^{2}}{5}+\dfrac{y^{2}}{b^{2}}=1 \) is:

If the equation of the hyperbola with foci \( (4,2) \) and \( (8,2) \) is \( 3x^{2} - y^{2} - \alpha x + \beta y + \gamma = 0 \), then \( \alpha + \beta + \gamma \) is equal to __________.

Consider a hyperbola \( H \) having center at the origin and foci on the \( x \)-axis. Let \( C_{1} \) be the circle touching the hyperbola \( H \) and having the centre at the origin. Let \( C_{2} \) be the circle touching the hyperbola \( H \) at its vertex and having the centre at one of its foci. If areas (in sq unit) of \( C_{1} \) and \( C_{2} \) are \( 36\pi \) and \( 4\pi \), respectively, then the length (in units) of latus rectum of \( H \) is:

Let \( E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1,\, a > b \) and \( H : \frac{x^{2}}{A^{2}} - \frac{y^{2}}{B^{2}} = 1 \). Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to:

Let \( x = 2t,\, y = \frac{t^2}{3} \) be a conic. Let \( S \) be the focus and \( B \) be the point on the axis of the conic such that \( SA \perp BA \), where \( A \) is any point on the conic. If \( k \) is the ordinate of the centroid of \( \triangle SAB \), then \( \lim\limits_{t \to 1} k \) is equal to:

If \( x^2 - y^2 + 2hxy + 2gx + 2fy + c = 0 \) is the locus of a point, which moves such that it is always equidistant from the lines \( x + 2y + 7 = 0 \) and \( 2x - y + 8 = 0 \), then the value of \( g + c + h - f \) equals.

Let the eccentricity of an ellipse \( \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1 \) is reciprocal to that of the hyperbola \( 2x^{2}-2y^{2}=1 \). If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse ________ .

The normal to the hyperbola \( \frac{x^2}{a^2}-\frac{y^2}{9}=1 \) at the point \( (8,\,3\sqrt{3}) \) on it passes through the point:

Let \( P_{1} \) be parabola with vertex \( (3, 2) \) and focus \( (4, 4) \) and \( P_{2} \) be its mirror image with respect to the line \( x + 2y = 6 \). Then the directrix of \( P_{2} \) is \( x + 2y = \) ________ .

Let \( T \) be the tangent to the ellipse \( E:\; x^{2}+4y^{2}=5 \) at the point \( P(1,1) \). If the area of the region bounded by the tangent \( T \), ellipse \( E \), lines \( x=1 \) and \( x=\sqrt{5} \) is \( \alpha \sqrt{5}+\beta+\gamma \cos ^{-1}\!\left(\dfrac{1}{\sqrt{5}}\right) \), then \( \left|\alpha+\beta+\gamma\right| \) is equal to _______.

Let the product of the focal distances of the point \( (\sqrt{3}, \tfrac{1}{2}) \) on the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( (a > b) \), be \( \tfrac{7}{4} \). Then the absolute difference of the eccentricities of two such ellipses is:

Let the foci of a hyperbola coincide with the foci of the ellipse \( \frac{x^2}{36} + \frac{y^2}{16} = 1 \). If the eccentricity of the hyperbola is \( 5 \), then the length of its latus rectum is:

Let \( H_{n}:\ \dfrac{x^{2}}{1+n}-\dfrac{y^{2}}{3+n}=1,\ n\in\mathbb{N} \). Let \( k \) be the smallest even value of \( n \) such that the eccentricity of \( H_{k} \) is a rational number. If \( l \) is the length of the latus rectum of \( H_{k} \), then \( 21l \) is equal to __________ .

Let \( H \) be the hyperbola, whose foci are \( (1 \pm \sqrt{2}, 0) \) and eccentricity is \( \sqrt{2} \). Then the length of its latus rectum is:

If \( x^{2} + 9y^{2} - 4x + 3 = 0, \, x, y \in \mathbb{R} \), then \( x \) and \( y \) respectively lie in the intervals:

The distance of the point \( (6,-2\sqrt{2}) \) from the common tangent \( y=mx+c,\, m>0 \), of the curves \( x=2y^{2} \) and \( x=1+y^{2} \) is:

Let the point \( P \) of the focal chord \( PQ \) of the parabola \( y^{2}=16x \) be \( (1,-4) \). If the focus of the parabola divides the chord \( PQ \) in the ratio \( m:n \), \( \gcd(m,n)=1 \), then \( m^{2}+n^{2} \) is equal to:

If the chord joining the points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ on the parabola $y^2 = 12x$ subtends a right angle at the vertex of the parabola, then $x_1x_2 - y_1y_2$ is equal to:

A line passing through the point \( P(\sqrt{5}, \sqrt{5}) \) intersects the ellipse \( \frac{x^2}{36} + \frac{y^2}{25} = 1 \) at \( A \) and \( B \) such that \( (PA)\cdot(PB) \) is maximum. Then \( 5(PA^2 + PB^2) \) is equal to:

Let a line \( L_1 \) be tangent to the hyperbola \( \dfrac{x^2}{16} - \dfrac{y^2}{4} = 1 \) and let \( L_2 \) be the line passing through the origin and perpendicular to \( L_1 \). If the locus of the point of intersection of \( L_1 \) and \( L_2 \) is \( (x^2 + y^2)^2 = \alpha x^2 + \beta y^2 \), then \( \alpha + \beta \) is equal to _______.

If the locus of the point, whose distances from the point \( (2, 1) \) and \( (1, 3) \) are in the ratio \( 5 : 4 \), is \( ax^{2} + by^{2} + cxy + dx + ey + 170 = 0 \), then the value of \( a^{2} + 2b + 3c + 4d + e \) is equal to:

Let the focal chord \( PQ \) of the parabola \( y^{2} = 4x \) make an angle of \( 60^\circ \) with the positive \( x \)-axis, where \( P \) lies in the first quadrant. If the circle, whose one diameter is \( PS \), \( S \) being the focus of the parabola, touches the \( y \)-axis at the point \( (0, \alpha) \), then \( 5\alpha^{2} \) is equal to:

Let the point \( P \) be the vertex of the parabola \( y = x^{2} - 6x + 12 \). If a line passing through the point \( P \) intersects the circle \( x^{2} + y^{2} - 2x - 4y + 3 = 0 \) at the point \( R \) and \( S \), then the maximum value of \( (PR + PS)^{2} \) is:

Let \( H:\; \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \) be a hyperbola such that the distance between its foci is \( 6 \) and the distance between its directrices is \( \frac{8}{3} \). If the line \( x=\alpha \) intersects the hyperbola H at the points \( A \) and \( B \) such that the area of the triangle \( AOB \) is \( 4\sqrt{15} \), where \( O \) is the origin, then \( \alpha^2 \) equals:

A tangent line \( L \) is drawn at the point \( (2,-4) \) on the parabola \( y^{2}=8x \). If the line \( L \) is also tangent to the circle \( x^{2}+y^{2}=a \), then ‘\( a \)’ is equal to ______.

Let the normal at the point \( P \) on the parabola \( y^2 = 6x \) pass through the point \( (5, -8) \). If the tangent at \( P \) to the parabola intersects its directrix at the point \( Q \), then the ordinate of the point \( Q \) is:

Let the latus rectum of the hyperbola \( \dfrac{x^2}{9} - \dfrac{y^2}{b^2} = 1 \) subtend an angle of \( \dfrac{\pi}{3} \) at the centre of the hyperbola. If \( b^2 \) is equal to \( \dfrac{l}{m}(1 + \sqrt{n}) \), where \( l \) and \( m \) are co-prime numbers, then \( l^2 + m^2 + n^2 \) is equal to \( \_\_\_\_\_\_\_\_ \).

Let \( O \) be the origin, and \( P \) and \( Q \) be two points on the rectangular hyperbola \( xy = 12 \) such that the mid-point of the line segment \( PQ \) is \( \left( \frac{1}{2}, -\frac{1}{2} \right) \). Then the area of the triangle \( OPQ \) equals:

Let \( C \) be the circle of minimum area enclosing the ellipse \( E:\; \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1 \) with eccentricity \( \dfrac{1}{2} \) and foci \( (\pm 2,0) \). Let \( PQR \) be a variable triangle, whose vertex \( P \) is on the circle \( C \) and the side \( QR \) of length \( 2a \) is parallel to the major axis of \( E \) and contains the point of intersection of \( E \) with the negative \( y \)-axis. Then the maximum area of the triangle \( PQR \) is:

If the radius of the largest circle with centre \( (2,0) \) inscribed in the ellipse \( x^2 + 4y^2 = 36 \) is \( r \), then \( 12r^2 \) is equal to:

Let the tangent drawn to the parabola \( y^2 = 24x \) at the point \( (\alpha, \beta) \) is perpendicular to the line \( 2x + 2y = 5 \). Then the normal to the hyperbola \( \dfrac{x^2}{\alpha^2} - \dfrac{y^2}{\beta^2} = 1 \) at the point \( (\alpha + 4, \beta + 4) \) does NOT pass through the point:

Let \( A \) be the point \((3, 0)\) and circles with variable diameter \( AB \) touch the circle \( x^2 + y^2 = 36 \) internally. Let the curve \( C \) be the locus of the point \( B \). If the eccentricity of \( C \) is \( e \), then \( 72e^2 \) is equal to __________.

Let \( ABCD \) be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides \( AD \) and \( BC \) of the trapezium be parallel to \( y \)-axis. If the diagonal \( AC \) is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of \( ABCD \) is:

If the \( x \)-intercept of a focal chord of the parabola \( y^2 = 8x + 4y + 4 \) is \( 3 \), then the length of this chord is equal to \( \_\_\_\_ \).

The equations of two sides of a variable triangle are \( x=0 \) and \( y=3 \), and its third side is a tangent to the parabola \( y^{2}=6x \). The locus of its circumcentre is:

Let \( O \) be the vertex of the parabola \( x^2 = 4y \) and \( Q \) be any point on it. Let the locus of the point \( P \), which divides the line segment \( OQ \) internally in the ratio \( 2:3 \) be the conic \( C \). Then the equation of the chord of \( C \), which is bisected at the point \( (1, 2) \) is:

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is:

Let each of the two ellipses \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, (a > b) \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, (A < B) \) have eccentricity \( \frac{4}{5} \). Let the lengths of the latus recta of \( E_1 \) and \( E_2 \) be \( l_1 \) and \( l_2 \), respectively, such that \( 2l_1^2 = 9l_2 \). If the distance between the foci of \( E_1 \) is \( 8 \), then the distance between the foci of \( E_2 \) is:

Let the line \( L:\ \sqrt{2}x+y=\alpha \) pass through the point of the intersection \( P \) (in the first quadrant) of the circle \( x^2+y^2=3 \) and the parabola \( x^2=2y \). Let the line \( L \) touch two circles \( C_1 \) and \( C_2 \) of equal radius \( 2\sqrt{3} \). If the centres \( Q_1 \) and \( Q_2 \) of the circles \( C_1 \) and \( C_2 \) lie on the y-axis, then the square of the area of the triangle \( PQ_1Q_2 \) is equal to ______.

If the eccentricity \( e \) of the hyperbola \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \), passing through \( (6, 4\sqrt{3}) \), satisfies \( 15(e^{2} + 1) = 34e \), then the length of the latus rectum of the hyperbola \( \frac{x^{2}}{b^{2}} - \frac{y^{2}}{2(a^{2} + 1)} = 1 \) is:

Let a line \( L: 2x + y = k,\ k > 0 \) be a tangent to the hyperbola \( x^2 - y^2 = 3 \). If \( L \) is also a tangent to the parabola \( y^2 = \alpha x \), then \( \alpha \) is equal to:

Let the foci of a hyperbola be \( (1, 14) \) and \( (1, -12) \). If it passes through the point \( (1, 6) \), then the length of its latus-rectum is:

For real numbers \( a, b \) (\( a > b > 0 \)), let
\( \text{Area}\left\{(x,y): x^2 + y^2 \le a^2 \text{ and } \frac{x^2}{a^2} + \frac{y^2}{b^2} \ge 1 \right\} = 30\pi \) and \( \text{Area}\left\{(x,y): x^2 + y^2 \ge b^2 \text{ and } \frac{x^2}{a^2} + \frac{y^2}{b^2} \le 1 \right\} = 18\pi \).
Then the value of \( (a-b)^2 \) is equal to ______.

Let \( A(0,1), B(1,1) \) and \( C(1,0) \) be the mid-points of the sides of a triangle with incentre at the point \( D \). If the focus of the parabola \( y^{2}=4ax \) passing through \( D \) is \( (\alpha+\beta\sqrt{2},0) \), where \( \alpha \) and \( \beta \) are rational numbers, then \( \dfrac{\alpha}{\beta^{2}} \) is equal to:

A common tangent \( T \) to the curves \( C_{1}:\ \dfrac{x^{2}}{4}+\dfrac{y^{2}}{9}=1 \) and \( C_{2}:\ \dfrac{x^{2}}{42}-\dfrac{y^{2}}{143}=1 \) does not pass through the fourth quadrant. If \( T \) touches \( C_{1} \) at \( (x_{1},y_{1}) \) and \( C_{2} \) at \( (x_{2},y_{2}) \), then \( |2x_{1}+x_{2}| \) is equal to ______.

Let \( P\left(\dfrac{2\sqrt{3}}{\sqrt{7}}, \dfrac{6}{\sqrt{7}}\right), Q, R \) and \( S \) be four points on the ellipse \( 9x^2 + 4y^2 = 36 \). Let \( PQ \) and \( RS \) be mutually perpendicular and pass through the origin. If \( \dfrac{1}{(PQ)^2} + \dfrac{1}{(RS)^2} = \dfrac{p}{q} \), where \( p \) and \( q \) are coprime, then \( p+q \) is equal to:

If the length of the latus rectum of a parabola, whose focus is \( (a,a) \) and the tangent at its vertex is \( x+y=a \), is \( 16 \), then \( |a| \) is equal to:

Let \( E \) be an ellipse whose axes are parallel to the co-ordinates axes, having its centre at \( (3,-4) \), one focus at \( (4,-4) \) and one vertex at \( (5,-4) \). If \( mx-y=4, \ m>0 \) is a tangent to the ellipse \( E \), then the value of \( 5m^{2} \) is equal to ____________.

Consider the parabola with vertex \( \left( \frac{1}{2}, \frac{3}{4} \right) \) and the directrix \( y = \frac{1}{2} \). Let \( P \) be the point where the parabola meets the line \( x = -\frac{1}{2} \). If the normal to the parabola at \( P \) intersects the parabola again in the point \( Q \), then \( (PQ)^2 \) is equal to:

For some \( \theta \in \left(0, \frac{\pi}{2}\right) \), let the eccentricity and the length of the latus rectum of the hyperbola \( x^2 - y^2 \sec^2 \theta = 8 \) be \( e_1 \) and \( l_1 \), respectively, and let eccentricity and the length of the latus rectum of the ellipse \( x^2 \sec^2 \theta + y^2 = 6 \) be \( e_2 \) and \( l_2 \), respectively. If \( e_1^2 = e_2^2 \left(\sec^2 \theta + 1\right) \), then \( \left|\dfrac{l_1 l_2}{e_1 e_2}\right| \tan^2 \theta \) is equal to __________.

Let the tangent to the parabola \( y^{2}=12x \) at the point \( (3,\alpha) \) be perpendicular to the line \( 2x+2y=3 \). Then the square of distance of the point \( (6,-4) \) from the normal to the hyperbola \( \alpha^{2}x^{2}-9y^{2}=9\alpha^{2} \) at its point \( (\alpha-1,\alpha+2) \) is equal to ______.

For \( 0<\theta<\frac{\pi}{2} \), if the eccentricity of the hyperbola \( x^2-y^2cosec^2\theta=5 \) is \( \sqrt{7} \) times eccentricity of the ellipse \( x^2cosec^2\theta+y^2=5 \), then the value of \( \theta \) is:

Let the line \( 2x+3y-k=0,\,k>0 \), intersect the \( x \)-axis and \( y \)-axis at the points \( A \) and \( B \), respectively. If the equation of the circle having the line segment \( AB \) as a diameter is \( x^{2}+y^{2}-3x-2y=0 \) and the length of the latus rectum of the ellipse \( x^{2}+9y^{2}=k^{2} \) is \( \frac{m}{n} \), where \( m \) and \( n \) are coprime, then \( 2m+n \) is equal to:

If the line \( 3x - 2y + 12 = 0 \) intersects the parabola \( 4y = 3x^2 \) at the points \( A \) and \( B \), then at the vertex of the parabola, the line segment \( AB \) subtends an angle equal to:

Let the length of the focal chord \( PQ \) of the parabola \( y^2 = 12x \) be \( 15 \) units. If the distance of \( PQ \) from the origin is \( p \), then \( 10p^2 \) is equal to \( \_\_\_\_\_\_\_\_\_\_ \).

The line \( x = 8 \) is the directrix of the ellipse \( E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) with the corresponding focus \( (2, 0) \). If the tangent to \( E \) at the point \( P \) in the first quadrant passes through the point \( (0, 4\sqrt{3}) \) and intersects the \( x \)-axis at \( Q \), then \( (3PQ)^{2} \) is equal to ______.

Let the tangent and normal at the point \( (3\sqrt{3}, 1) \) on the ellipse \( \dfrac{x^2}{36} + \dfrac{y^2}{4} = 1 \) meet the y-axis at the points \( A \) and \( B \) respectively. Let the circle \( C \) be drawn taking \( AB \) as a diameter and the line \( x = 2\sqrt{5} \) intersect \( C \) at the points \( P \) and \( Q \). If the tangents at the points \( P \) and \( Q \) on the circle intersect at the point \( (\alpha, \beta) \), then \( \alpha^2 - \beta^2 \) is equal to:

Let the image of parabola \( x^{2}=4y \) in the line \( x-y=1 \) be \( (y+a)^{2}=b(x-c) \), \( a,b,c \in \mathbb{N} \). Then \( a+b+c \) is equal to:

If two tangents drawn from a point \( P \) to the parabola \( y^{2}=16(x-3) \) are at right angles, then the locus of point \( P \) is:

The vertices of a hyperbola \( H \) are \( (\pm 6, 0) \) and its eccentricity is \( \frac{\sqrt{5}}{2} \). Let \( N \) be the normal to \( H \) at a point in the first quadrant and parallel to the line \( \sqrt{2}x + y = 2\sqrt{2} \). If \( d \) is the length of the line segment of \( N \) between \( H \) and the \( y \)-axis then \( d^2 \) is equal to ______ .

Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1,\, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be \( 4 \). If \( E_1 \) and \( E_2 \) meet at \( A \), \( B \), \( C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals:

Let the ellipse $3x^2 + py^2 = 4$ pass through the centre $C$ of the circle $x^2 + y^2 - 2x - 4y - 11 = 0$ of radius $r$. Let $f_1, f_2$ be the focal distances of the point $C$ on the ellipse. Then $6 f_1 f_2 - r$ is equal to:

Let \( O \) be the vertex of the parabola \( y^{2}=4x \) and its chords \( OP \) and \( OQ \) are perpendicular to each other. If the locus of the mid-point of the line segment \( PQ \) is a conic \( C \), then the length of its latus rectum is:

In a group of \( 100 \) persons \( 75 \) speak English and \( 40 \) speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is \( \alpha \) and the number of persons who speak only Hindi is \( \beta \), then the eccentricity of the ellipse \( 25(\beta^{2}x^{2}+a^{2}y^{2})=\alpha^{2}\beta^{2} \) is:

If \( S \) and \( S' \) are the foci of the ellipse \( \frac{x^{2}}{18} + \frac{y^{2}}{9} = 1 \) and \( P \) be a point on the ellipse, then \( \min(SP \cdot S'P) + \max(SP \cdot S'P) \) is equal to:

Let the shortest distance from \( (a, 0), a > 0 \), to the parabola \( y^{2} = 4x \) be \( 4 \). Then the equation of the circle passing through the point \( (a, 0) \) and the focus of the parabola, the having its centre on the axis of the parabola is:

Let one focus of the hyperbola \( H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) be at \( (\sqrt{10}, 0) \) and the corresponding directrix be \( x = \frac{9}{\sqrt{10}} \). If \( e \) and \( l \) respectively are the eccentricity and the length of the latus rectum of \( H \), then \( 9(e^2 + l) \) is equal to:

Let the eccentricity of the hyperbola \( H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) be \( \sqrt{\frac{5}{2}} \) and length of its latus rectum be \( 6\sqrt{2} \). If \( y = 2x + c \) is a tangent to the hyperbola \( H \), then the value of \( c^2 \) is equal to:

Let \( A (\sec \theta, 2 \tan \theta) \) and \( B (\sec \phi, 2 \tan \phi) \), where \( \theta + \phi = \frac{\pi}{2} \), be two points on the hyperbola \( 2x^{2} - y^{2} = 2 \). If \( (\alpha, \beta) \) is the point of the intersection of the normals to the hyperbola at \( A \) and \( B \), then \( (2\beta)^{2} \) is equal to ________.

Let \( L_1, L_2 \) be the lines passing through the point \( P(0,1) \) and touching the parabola \( 9x^2 + 12x + 18y - 14 = 0 \). Let \( Q \) and \( R \) be the points on the lines \( L_1 \) and \( L_2 \) such that the \( \triangle PQR \) is an isosceles triangle with base \( QR \). If the slopes of the lines \( QR \) are \( m_1 \) and \( m_2 \), then \( 16\left(m_1^2 + m_2^2\right) \) is equal to ____.

Let the common tangents to the curves \( 4(x^{2}+y^{2}) = 9 \) and \( y^{2} = 4x \) intersect at the point \( Q \). Let an ellipse, centered at the origin \( O \), has lengths of semi-minor and semi-major axes equal to \( OQ \) and \( 6 \), respectively. If \( e \) and \( l \) respectively denote the eccentricity and the length of the latus rectum of this ellipse, then \( \frac{l}{e^{2}} \) is equal to______________.

Let \( PQ \) be a chord of the hyperbola \( \frac{x^2}{4}-\frac{y^2}{b^2}=1 \), perpendicular to the \( x \)-axis such that \( OPQ \) is an equilateral triangle, \( O \) being the centre of the hyperbola. If the eccentricity of the hyperbola is \( \sqrt{3} \), then the area of the triangle \( OPQ \) is:

Let \( PQ \) be a chord of the parabola \( y^2 = 12x \) and the midpoint of \( PQ \) be at \( (4, 1) \). Then, which of the following point lies on the line passing through the points \( P \) and \( Q \)?

If two tangents drawn from a point \( (\alpha, \beta) \) lying on the ellipse \( 25x^{2} + 4y^{2} = 1 \) to the parabola \( y^{2} = 4x \) are such that the slope of one tangent is four times the other, then the value of \( (10\alpha + 5)^{2} + (16\beta^{2} + 50)^{2} \) equals ________________.

If the tangent at a point \( P \) on the parabola \( y^{2}=3x \) is parallel to the line \( x+2y=1 \) and the tangents at the points \( Q \) and \( R \) on the ellipse \( \dfrac{x^{2}}{4}+\dfrac{y^{2}}{1}=1 \) are perpendicular to the line \( x-y=2 \), then the area of the triangle \( PQR \) is:

Let \( T \) and \( C \) respectively be the transverse and conjugate axes of the hyperbola \( 16x^{2}-y^{2}+64x+4y+44=0 \). Then the area of the region above the parabola \( x^{2}=y+4 \), below the transverse axis \( T \) and on the right of the conjugate axis \( C \) is:

Let the tangents at the points \( P \) and \( Q \) on the ellipse \( \dfrac{x^{2}}{2} + \dfrac{y^{2}}{4} = 1 \) meet at the point \( R(\sqrt{2},\, 2\sqrt{2} - 2) \). If \( S \) is the focus of the ellipse on its negative major axis, then \( SP^{2} + SQ^{2} \) is equal to __________________.

Consider the circle \( C: x^2 + y^2 = 4 \) and the parabola \( P: y^2 = 8x \). If the set of all values of \( \alpha \), for which three chords of the circle \( C \) on three distinct lines passing through the point \( (\alpha, 0) \) are bisected by the parabola \( P \) is the interval \( (p, q) \), then \( (2q - p)^2 \) is equal to ________.

Let \( R \) be the focus of the parabola \( y^2 = 20x \) and the line \( y = mx + c \) intersect the parabola at two points \( P \) and \( Q \). Let the point \( G(10, 10) \) be the centroid of the triangle \( PQR \). If \( c - m = 6 \), then \( (PQ)^2 \) is:

Let \( e_{1} \) and \( e_{2} \) be two distinct roots of the equation \( x^{2}-ax+2=0 \). Let the sets \( \{\, a \in \mathbb{R} : e_{1} \) and \( e_{2} \) are the eccentricities of hyperbolas \} \( = (\alpha,\beta) \), and \( \{\, a \in \mathbb{R} : e_{1} \) and \( e_{2} \) are the eccentricities of an ellipse and a hyperbola, respectively \} \( = (\gamma,\infty) \).
Then \( \alpha^{2}+\beta^{2}+\gamma^{2} \) is equal to:

An ellipse \( E:\; \displaystyle\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) passes through the vertices of the hyperbola \( H:\; \displaystyle\frac{x^{2}}{49}-\frac{y^{2}}{64}=-1 \). Let the major and minor axes of the ellipse \( E \) coincide with the transverse and conjugate axes of the hyperbola \( H \), respectively. Let the product of the eccentricities of \( E \) and \( H \) be \( {1}/{2} \). If \( l \) is the length of the latus rectum of the ellipse \( E \), then the value of \( 113l \) is equal to ______.

The length of the latus rectum of a parabola, whose vertex and focus are on the positive \( x \)-axis at a distance \( R \) and \( S \) (\( > R \)) respectively from the origin, is:

The foci of a hyperbola are \( (\pm 2, 0) \) and its eccentricity is \( \dfrac{3}{2} \). A tangent, perpendicular to the line \( 2x + 3y = 6 \), is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the x and y-axes are \( a \) and \( b \) respectively, then \( |6a| + |5b| \) is equal to _________.

The length of the latus rectum and directrices of a hyperbola with eccentricity \( e \) are \( 9 \) and \( x = \pm \frac{4}{\sqrt{3}} \), respectively. Let the line \( y-\sqrt{3}x+\sqrt{3}=0 \) touch this hyperbola at \( (x_0,y_0) \). If \( m \) is the product of the focal distances of the point \( (x_0,y_0) \), then \( 4e^2+m \) is equal to ______.

Let \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \), \( a>b \) be an ellipse, whose eccentricity is \( \frac{1}{\sqrt{2}} \) and the length of the latusrectum is \( \sqrt{14} \). Then the square of the eccentricity of \( \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) is:

The focus of the parabola \( y^2 = 4x + 16 \) is the centre of the circle \( C \) of radius \( 5 \). If the value of \( \lambda \), for which \( C \) passes through the point of intersection of the lines \( 3x - y = 0 \) and \( x + \lambda y = 4 \), are \( \lambda_1 \) and \( \lambda_2 \), \( \lambda_1 < \lambda_2 \), then \( 12\lambda_1 + 29\lambda_2 \) is equal to __________.

The line \( 12x \cos \theta + 5y \sin \theta = 60 \) is tangent to which of the following curves?

Let \( P(a,b) \) be a point on the parabola \( y^{2}=8x \) such that the tangent at \( P \) passes through the centre of the circle \( x^{2}+y^{2}-10x-14y+65=0 \). Let \( A \) be the product of all possible values of \( a \) and \( B \) be the product of all possible values of \( b \). Then the value of \( A+B \) is equal to:

Let a circle of radius \( 4 \) be concentric to the ellipse \( 15x^2+19y^2=285 \). Then the common tangents are inclined to the minor axis of the ellipse at the angle.

If the points of intersection of the ellipse \( \dfrac{x^2}{16} + \dfrac{y^2}{b^2} = 1 \) and the circle \( x^2 + y^2 = 4b,\, b>4 \) lie on the curve \( y^2 = 3x^2 \), then \( b \) is equal to:

The length of the chord of the ellipse \( \dfrac{x^2}{25} + \dfrac{y^2}{16} = 1 \), whose mid-point is \( \left( 1, \dfrac{2}{5} \right) \), is equal to:

Let \( A \), \( B \) and \( C \) be three points on the parabola \( y^2 = 6x \) and let the line segment \( AB \) meet the line \( L \) through \( C \) parallel to the \( x \)-axis at the point \( D \). Let \( M \) and \( N \) respectively be the feet of perpendiculars from \( A \) and \( B \) on \( L \). Then \( \left(\frac{AM \cdot BN}{CD}\right)^2 \) is equal to ________.

A circle with centre \( (2, 3) \) and radius \( 4 \) intersects the line \( x + y = 3 \) at the points \( P \) and \( Q \). If the tangents at \( P \) and \( Q \) intersect at the point \( S(\alpha, \beta) \), then \( 4\alpha - 7\beta \) is equal to _________.

Let a tangent to the curve \( y^{2}=24x \) meet the curve \( xy=2 \) at the points \( A \) and \( B \). Then the mid-points of such line segment \( AB \) lie on a parabola with the:

Let the directrix of the parabola \( P: y^{2} = 8x \), cut x-axis at the point \( A \). Let \( B(\alpha, \beta),\ \alpha > 1 \), be a point on \( P \) such that the slope of \( AB \) is \( \frac{3}{5} \). If \( BC \) is a focal chord of \( P \), then six times the area of \( \triangle ABC \) is:

If \(P\) is a point on the parabola \(y=x^{2}+4\) which is closest to the straight line \(y=4x-1\), then the co-ordinates of \(P\) are:

Let \( E_1 : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1, \ a > b. \) Let \( E_2 \) be another ellipse such that it touches the end points of major axis of \( E_1 \) and the foci of \( E_2 \) are the end points of minor axis of \( E_1 \). If \( E_1 \) and \( E_2 \) have same eccentricities, then its value is:

Let \( A \), \( B \) and \( C \) be the vertices of a variable right angled triangle inscribed in the parabola \( y^2 = 16x \). Let the vertex \( B \) containing the right angle be \( (4, 8) \) and the locus of the centroid of \( \Delta ABC \) be a conic \( C_0 \). Then three times the length of latus rectum of \( C_0 \) is ________.

The locus of the centroid of the triangle formed by any point \( P \) on the hyperbola \( 16x^{2}-9y^{2}+32x+36y-164=0 \) and its foci is:

If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9} + \frac{y^2}{25} = 1$ and eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt{2}, \frac{14}{3}\sqrt{\frac{2}{5}}\right)$ on the hyperbola, is equal to:

The tangents at the points \( A(1, 3) \) and \( B(1, -1) \) on the parabola \( y^2 - 2x - 2y = 1 \) meet at the point \( P \). Then the area (in \( \text{unit}^2 \)) of the triangle \( PAB \) is:

Let the length of the latus rectum of an ellipse \( \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ (a>b), \) be \( 30 \). If its eccentricity is the maximum value of the function \( f(t)=-\dfrac{3}{4}+2t-t^{2} \), then \( \left(a^{2}+b^{2}\right) \) is equal to:

Let \( L \) be a tangent to the parabola \( y^{2} = 4x - 20 \) at \( (6, 2) \). If \( L \) is also a tangent to the ellipse \( \dfrac{x^{2}}{2} + \dfrac{y^{2}}{b} = 1 \), then the value of \( b \) is equal to:

The axis of a parabola is the line \( y = x \) and its vertex and focus are in the first quadrant at distances \( \sqrt{2} \) and \( 2\sqrt{2} \) units from the origin, respectively. If the point \( (1,k) \) lies on the parabola, then a possible value of \( k \) is:

Consider a hyperbola \( H : {x^{2}} - {2y^{2}} = 4 \). Let the tangent at a point \( P(4,\sqrt{6}) \) meet the x-axis at \( Q \) and latus rectum at \( R(x_{1},y_{1}), x_{1} > 0 \). If \( F \) is a focus of \( H \) which is nearer to the point \( P \), then the area of \( \triangle QFR \) is equal to:

Let \( f(x)=x^{2}+9 \), \( g(x)=\dfrac{x}{x-9} \) and \( a=f o g(10),\, b=g o f(3) \). If \( e \) and \( l \) denote the eccentricity and the length of the latus rectum of the ellipse \( \dfrac{x^{2}}{a}+\dfrac{y^{2}}{b}=1 \), then \( 8e^{2}+l^{2} \) is equal to:

Let an ellipse \( E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ a^{2}>b^{2} \), passes through \( \left(\dfrac{\sqrt{3}}{2},1\right) \) and has eccentricity \( \dfrac{1}{\sqrt{3}} \). If a circle, centered at focus \( F(\alpha,0),\ \alpha>0 \) of \( E \) and radius \( \dfrac{2}{\sqrt{3}} \) intersects \( E \) at two points \( P \) and \( Q \), then \( PQ^{2} \) is equal to:

Let the line \( y - x = 1 \) intersect the ellipse \( \frac{x^{2}}{2} + \frac{y^{2}}{1} = 1 \) at the points \( A \) and \( B \). Then the angle made by the line segment \( AB \) at the center of the ellipse is:

Two tangent lines \(l_1\) and \(l_2\) are drawn from the point \( (2, 0) \) to the parabola \( 2y^2 = -x \). If the lines \(l_1\) and \(l_2\) are also tangent to the circle \( (x-5)^2 + y^2 = r \), then \( 17r \) is equal to _______________.

Let \( P \) be a point on the ellipse \(\displaystyle \frac{x^2}{9} + \frac{y^2}{4} = 1 \). Let the line passing through \( P \) and parallel to \( y \)-axis meet the circle \( x^2 + y^2 = 9 \) at point \( Q \) such that \( P \) and \( Q \) are on the same side of the \( x \)-axis. Then, the eccentricity of the locus of the point \( R \) on \( PQ \) such that \( PR:RQ = 4:3 \) as \( P \) moves on the ellipse, is:

The radius of the smallest circle which touches the parabolas \( y = x^{2} + 2 \) and \( x = y^{2} + 2 \) is:

If \( m \) is the slope of a common tangent to the curves \( \dfrac{x^{2}}{16} + \dfrac{y^{2}}{9} = 1 \) and \( x^{2} + y^{2} = 12 \), then \( 12 m^{2} \) is equal to:

The area of the region enclosed by the curves \( y = x^{2} - 4x + 4 \) and \( y^{2} = 16 - 8x \) is:

Let the hyperbola \( H : \frac{x^{2}}{a^{2}} - y^{2} = 1 \) and the ellipse \( E : 3x^{2} + 4y^{2} = 12 \) be such that the length of latus rectum of \( H \) is equal to the length of latus rectum of \( E \). If \( e_{H} \) and \( e_{E} \) are the eccentricities of \( H \) and \( E \) respectively, then the value of \( 12\left(e_{H}^{2} + e_{E}^{2}\right) \) is equal to ________ .

Let \( P \) be a moving point on the circle \( x^2 + y^2 - 6x - 8y + 21 = 0 \). Then, the maximum distance of \( P \) from the vertex of the parabola \( x^2 + 6x + y + 13 = 0 \) is equal to:

If the equation of the parabola, whose vertex is at \( (5,4) \) and the directrix is \( 3x + y - 29 = 0 \), is \( x^{2} + a y^{2} + bxy + cx + dy + k = 0 \), then \( a + b + c + d + k \) is equal to

The area (in sq. units) of the region enclosed between the parabola \( y^{2} = 2x \) and the line \( x + y = 4 \) is ________ .

Consider the sets \( A = \{(x,y) \in \mathbb{R} \times \mathbb{R} : x^{2} + y^{2} = 25\} \), \( B = \{(x,y) \in \mathbb{R} \times \mathbb{R} : x^{2} + 9y^{2} = 144\} \), \( C = \{(x,y) \in \mathbb{Z} \times \mathbb{Z} : x^{2} + y^{2} \le 4\} \) and \( D = A \cap B \). The total number of one-one functions from the set \( D \) to the set \( C \) is:

If the maximum distance of normal to the ellipse \( \frac{x^2}{4} + \frac{y^2}{b^2} = 1, \; b < 2 \), from the origin is \( 1 \), then the eccentricity of the ellipse is:

If the line \(\alpha x+4y=\sqrt{7} \), where \( a\in \mathbb{R} \), touches the ellipse \( 3x^2+4y^2=1 \) at the point \( P \) in the first quadrant, then one of the focal distances of \( P \) is:

Let \( l \) be a line which is normal to the curve \( y = 2x^2 + x + 2 \) at a point \( P \) on the curve. If the point \( Q(6, 4) \) lies on the line \( l \) and \( O \) is origin, then the area of the triangle \( OPQ \) is equal to ________.

Let \( S \) be the focus of the hyperbola \( \dfrac{x^{2}}{3}-\dfrac{y^{2}}{5}=1 \), on the positive \( x \)-axis. Let \( C \) be the circle with its centre at \( A(\sqrt{6},\sqrt{5}) \) and passing through the point \( S \). If \( O \) is the origin and \( SAB \) is a diameter of \( C \), then the square of the area of the triangle \( OSB \) is equal to ________.

Let \( \lambda x - 2y = \mu \) be a tangent to the hyperbola \( a^{2}x^{2} - y^{2} = b^{2} \). Then \( \left( \frac{\lambda}{a} \right)^{2} - \left( \frac{\mu}{b} \right)^{2} \) is equal to:

For which of the following curves, the line \(x+\sqrt{3}\,y=2\sqrt{3}\) is the tangent at the point \(\left(\dfrac{3\sqrt{3}}{2},\dfrac{1}{2}\right)\)?

Let \( A(\alpha, 0) \) and \( B(0, \beta) \) be the points on the line. Let the point \( P \) divide the line segment \( AB \) internally in the ratio \( 7:3 \). Let \( 3x - 25 = 0 \) be a directrix of the ellipse \( E: \, \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \) and the corresponding focus be \( S \). If from \( S \), the perpendicular on the \( x \)-axis passes through \( P \), then the length of the latus rectum of \( E \) is equal to:

Let the foci of the ellipse \( \dfrac{x^{2}}{16} + \dfrac{y^{2}}{7} = 1 \) and the hyperbola \( \dfrac{x^{2}}{144} - \dfrac{y^{2}}{\alpha} = \dfrac{1}{25} \) coincide. Then the length of the latus rectum of the hyperbola is:

Let \( H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1 , \ a>0 , \ b>0 \) be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is \( 4\left(2\sqrt{2}+\sqrt{14}\right) \). If the eccentricity \( H \) is \( \dfrac{\sqrt{11}}{2} \), then the value of \( a^2+b^2 \) is equal to __________.

Let chord \( PQ \) of length \( 3\sqrt{13} \) of the parabola \( y^2 = 12x \) be such that the ordinates of points \( P \) and \( Q \) are in the ratio \( 1:2 \). If the chord \( PQ \) subtends an angle \( \alpha \) at the focus of the parabola, then \( \sin \alpha \) is equal to:

Let \( P(x_{0},y_{0}) \) be the point on the hyperbola \( 3x^{2}-4y^{2}=36 \), which is nearest to the line \( 3x+2y=1 \). Then \( \sqrt{2}\,(y_{0}-x_{0}) \) is equal to:

Let a tangent to the curve \( 9x^{2} + 16y^{2} = 144 \) intersect the coordinate axes at the points \( A \) and \( B \). Then, the minimum length of the line segment \( AB \) is _______.

A square \( ABCD \) has all its vertices on the curve \( x^2 y^2 = 1 \). The midpoints of its sides also lie on the same curve. Then, the square of area of \( ABCD \) is ______________ .

Let a parabola \( P \) be such that its vertex and focus lie on the positive \( x \)-axis at a distance \( 2 \) and \( 4 \) units from the origin, respectively. If tangents are drawn from \( O(0,0) \) to the parabola \( P \) which meet \( P \) at \( S \) and \( R \), then the area (in sq. units) of \( \triangle S O R \) is equal to:

If the shortest distance of the parabola \( y^2 = 4x \) from the centre of the circle \( x^2 + y^2 - 4x - 16y + 64 = 0 \) is \( d \), then \( d^2 \) is equal to:

Suppose \( AB \) is a focal chord of the parabola \( y^{2} = 12x \) of length \( l \) and slope \( m < \sqrt{3} \). If the distance of the chord \( AB \) from the origin is \( d \), then \( l d^{2} \) is equal to __________.

Let the hyperbola \( H: \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \) pass through the point \( (2\sqrt{2}, -2\sqrt{2}) \). A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is \( e \) times the length of the latus rectum of H, where \( e \) is the eccentricity of H, then which of the following points lies on the parabola

Let \( y=mx+c,\; m>0 \) be the focal chord of \( y^{2}=-64x \), which is tangent to \( (x+10)^{2}+y^{2}=4 \). Then, the value of \( 4\sqrt{2}\,(m+c) \) is equal to _______.

Let \( R \) be a rectangle given by the lines \( x=0 \), \( x=2 \), \( y=0 \) and \( y=5 \). Let \( A(\alpha, 0) \) and \( B(0, \beta) \), \( \alpha \in [0, 2] \) and \( \beta \in [0, 5] \), be such that the line segment \( AB \) divides the area of the rectangle \( R \) in the ratio \( 4:1 \). Then, the mid-point of \( AB \) lies on a :

The shortest distance between the curves \( y^2 = 8x \) and \( x^2 + y^2 + 12y + 35 = 0 \) is:

Let \( C \) be the largest circle centred at \( (2, 0) \) and inscribed in the ellipse \( \frac{x^{2}}{36} + \frac{y^{2}}{16} = 1 \). If \( (1, \alpha) \) lies on \( C \), then \( 10\alpha^{2} \) is equal to ______.

Let \( P: y^{2}=4ax,\; a>0 \) be a parabola with focus \( S \). Let the tangents to the parabola \( P \) make an angle of \( \frac{\pi}{4} \) with the line \( y=3x+5 \) touch the parabola \( P \) at \( A \) and \( B \). Then the value of \( a \) for which \( A \), \( B \) and \( S \) are collinear is:

Let \( P(4, 4\sqrt{3}) \) be a point on the parabola \( y^2 = 4ax \) and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:

If \( y=m_{1}x+c_{1} \) and \( y=m_{2}x+c_{2},\ m_{1}\neq m_{2} \) are two common tangents of circle \( x^{2}+y^{2}=2 \) and parabola \( y^{2}=x \), then the value of \( 8|\,m_{1}m_{2}\,| \) is equal to:

If the minimum area of the triangle formed by a tangent to the ellipse \( \dfrac{x^2}{b^2} + \dfrac{y^2}{4a^2} = 1 \) and the co-ordinate axis is \( kab \), then \( k \) is equal to \( \_\_\_\_\_\_\_\_\_\_ \).

The ordinates of the points \( P \) and \( Q \) on the parabola with focus \( (3,0) \) and directrix \( x=-3 \) are in the ratio \( 3:1 \). If \( R(\alpha,\beta) \) is the point of intersection of the tangents to the parabola of \( P \) and \( Q \), then \( \dfrac{\beta^2}{\alpha} \) is equal to ________.

Let the eccentricity of a hyperbola satisfy the equation \( 6e^{2} - 11e + 3 = 0 \). If the foci of the hyperbola are \( (3, 5) \) and \( (3, -4) \), then the length of its latus rectum is:

Two parabolas have the same focus \( (4,3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersects at the points \( A \) and \( B \), then \( (AB)^2 \) is equal to:

Let a tangent be drawn to the ellipse \( \dfrac{x^{2}}{27} + y^{2} = 1 \) at \( \left(3\sqrt{3}\cos\theta,\ \sin\theta\right) \) where \( \theta \in \left(0,\ \dfrac{\pi}{2}\right) \). Then the value of \( \theta \) such that the sum of intercepts on axes made by this tangent is minimum is equal to:

A tangent and a normal are drawn at the point \( P(2,\,-4) \) on the parabola \( y^2 = 8x \), which meet the directrix of the parabola at the points \( A \) and \( B \) respectively. If \( Q(a,\, b) \) is a point such that \( AQBP \) is square, then \( 2a + b \) is equal to:

Let the equation of two diameters of a circle \( x^2 + y^2 - 2x + 2fy + 1 = 0 \) be \( 2px - y = 1 \) and \( 2x + py = 4p \). Then the slope \( m \in (0,\infty) \) of the tangent to the hyperbola \( 3x^2 - y^2 = 3 \) passing through the centre of the circle is equal to ________.

Let an ellipse \( \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1 \), pass through the point (4, 3) and have eccentricity Then the length of its latus rectum is:

Let for two distinct values of \( p \) the lines \( y = x + p \) touch the ellipse E: \( \frac{x^2}{4^2} + \frac{y^2}{3^2} = 1 \) at the points \( A \) and \( B \). Let the line \( y = x \) intersect \( E \) at the points \( C \) and \( D \). Then the area of the quadrilateral \( ABCD \) is equal to:

Let the eccentricity of an ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( a > b \), be \( \frac{1}{4} \). If this ellipse passes through the point \( \left(-4\sqrt{\frac{2}{5}},\, 3\right) \), then \( a^2 + b^2 \) is equal to:

Let \( (h,k) \) lie on the circle \( C:x^{2}+y^{2}=4 \) and the point \( (2h+1,\,3k+2) \) lie on an ellipse with eccentricity \( e \). Then the value of \( \dfrac{5}{e^{2}} \) is equal to ____________.

If the area of the triangle formed by the positive \( x \)-axis, the normal and the tangent to then circle \( (x-2)^2 + (y-3)^2 = 25 \) at the point \( (5,7) \) is \( A \), then \( 24A \) is equal to __________.

The locus of mid-points of the line segments joining \( (-3,-5) \) and the points on the ellipse \( \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1 \) is:

Let \( H : \frac{-x^2}{a^2} + \frac{y^2}{b^2} = 1 \) be the hyperbola, whose eccentricity is \( \sqrt{3} \) and the length of the latus rectum is \( 4\sqrt{3} \). Suppose the point \( (\alpha, 6) \), \( \alpha > 0 \) lies on \( H \). If \( \beta \) is the product of the focal distances of the point \( (\alpha, 6) \), then \( \alpha^2 + \beta \) is equal to:

Let the foci of a hyperbola \( H \) coincide with the foci of the ellipse \(\displaystyle E:\ \frac{(x-1)^{2}}{100}+\frac{(y-1)^{2}}{75}=1 \) and the eccentricity of the hyperbola \( H \) be the reciprocal of the eccentricity of the ellipse \( E \). If the length of the transverse axis of \( H \) is \( \alpha \) and the length of its conjugate axis is \( \beta \), then \( 3\alpha^{2}+2\beta^{2} \) is equal to

Let \( PQ \) be a focal chord of the parabola \( y^{2} = 36x \) of length \( 100 \), making an acute angle with the positive \( x \)-axis. Let the ordinate of \( P \) be positive and \( M \) be the point on the line segment \( PQ \) such that \( PM : MQ = 3 : 1 \). Then which of the following points does NOT lie on the line passing through \( M \) and perpendicular to the line \( PQ \)?

Let \( a>0, b>0 \). Let \( e \) and \( l \) respectively be the eccentricity and length of the latus rectum of the hyperbola \( \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \). Let \( e' \) and \( l' \) respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If \( e^2 = \dfrac{11}{14}\,l \) and \( (e')^2 = \dfrac{11}{8}\,l' \), then the value of \( 77a + 44b \) is equal to:

The locus of the mid point of the line segment joining the point \( (4, 3) \) and the points on the ellipse \( x^2 + 2y^2 = 4 \) is an ellipse with eccentricity:

Let the mirror image of a circle \( c_1:x^2+y^2-2x-6y+\alpha=0 \) in line \( y=x+1 \) be \( c_2:5x^2+5y^2+10gx+10fy+38=0 \). If \( r \) is the radius of circle \( c_2 \), then \( \alpha+6r^2 \) is equal to__________.

Let \( C \) be the locus of the mirror image of a point on the parabola \( y^{2}=4x \) with respect to the line \( y=x \). Then the equation of tangent to \( C \) at \( P(2,1) \) is:

Let \( \theta \) be the acute angle between the tangents to the ellipse \( \frac{x^2}{9} + \frac{y^2}{1} = 1 \) and the circle \( x^2 + y^2 = 3 \) at their point of intersection in the first quadrant. Then \( \tan \theta \) is equal to:

The length of the chord of the ellipse \( \displaystyle\frac{x^2}{4} + \frac{y^2}{2} = 1 \), whose mid-point is \(\displaystyle \left( 1, \frac{1}{2} \right) \), is:

An equilateral triangle \( OAB \) is inscribed in the parabola \( y^2 = 4x \) with the vertex \( O \) at the vertex of the parabola. Then the minimum distance of the circle having \( AB \) as a diameter from the origin is:

If the points of intersection of two distinct conics \( x^2 + y^2 = 4b \) and \( \dfrac{x^2}{16} + \dfrac{y^2}{b^2} = 1 \) lie on the curve \( y^2 = 3x^2 \), then \( 3\sqrt{3} \) times the area of the rectangle formed by intersection points is __________.

Let the product of the focal distance of the point \( P(4, 2\sqrt{3}) \) on the hyperbola \( H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) be \( 32 \). Let the length of the conjugate axis of \( H \) be \( p \) and the length of its latus rectum be \( q \). Then \( p^2 + q^2 \) is equal to ________.

Let the maximum area of the triangle that can be inscribed in the ellipse \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{4} = 1 \), \( a > 2 \), having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the \( y \)-axis, be \( 6\sqrt{3} \). Then the eccentricity of the ellipse is:

$\text{Consider the hyperbola } \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text{ having one of its focus at } P(-3,0). \text{ If the latus}$ $\text{rectum through its other focus subtends a right angle at } P \text{ and } a^2b^2 = \alpha\sqrt{2} - \beta, \alpha, \beta \in \mathbb{N},$ $\text{then } \alpha + \beta \text{ is ____________ .}$

The line \( y = x + 1 \) meets the ellipse \( \frac{x^2}{4} + \frac{y^2}{2} = 1 \) at two points \( P \) and \( Q \). If \( r \) is the radius of the circle with \( PQ \) as diameter then \( (3r)^2 \) is equal to:

The eccentricity of an ellipse \( E \) with centre at the origin \( O \) is \( \dfrac{\sqrt{3}}{2} \) and its directrices are \( x = \pm \dfrac{4\sqrt{6}}{3} \). Let \( H : \dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1 \) be a hyperbola whose eccentricity is equal to the length of semi-major axis of \( E \), and whose length of latus rectum is equal to the length of minor axis of \( E \). Then the distance between the foci of \( H \) is:

Let the tangent to the parabola \( S: y^{2}=2x \) at the point \( P(2,2) \) meet the \( x \)-axis at \( Q \) and normal at it meet the parabola \( S \) at the point \( R \). Then the area (in sq. units) of the triangle \( PQR \) is equal to:

Consider the parabola \( P: y^{2}=4kx \) and the ellipse \( E: \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1 \). Let the line segment joining the points of intersection of \( P \) and \( E \), be their latus rectums. If the eccentricity of \( E \) is \( e \), then \( e^{2}+2\sqrt{2} \) is equal to ________.

A circle of radius \( 2 \) unit passes through the vertex and the focus of the parabola \( y^{2} = 2x \) and touches the parabola \( y = \left( x - \dfrac{1}{4} \right)^{2} + \alpha \), where \( \alpha > 0 \). Then \( (4\alpha - 8)^{2} \) is equal to _______.

Let \( S \) and \( S' \) be the foci of the ellipse \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \) and \( P(\alpha, \beta) \) be a point on the ellipse in the first quadrant. If \( (SP)^2 + (S'P)^2 - SP \cdot S'P = 37 \), then \( \alpha^2 + \beta^2 \) is equal to:

The equation of a common tangent to the parabolas \( y = x^2 \) and \( y = -(x-2)^2 \) is:

Let \( S=\{(x,y)\in \mathbb{N}\times \mathbb{N}:9(x-3)^2+16(y-4)^2\leq 144\} \) and \(\displaystyle T = \{(x, y) \in \mathbb{R} \times \mathbb{R} : (x - 7)^2 + (y - 4)^2 \le 36\}\). Then \(n(S\cap T)\) is equal to __________.

Let \( e_1 \) be the eccentricity of the hyperbola \(\displaystyle \frac{x^2}{16}-\frac{y^2}{9}=1 \) and \( e_2 \) be the eccentricity of the ellipse \(\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b \), which passes through the foci of the hyperbola. If \( e_1 e_2=1 \), then the length of the chord of the ellipse parallel to the \( x \)-axis and passing through \( (0,2) \) is:

If a tangent to the ellipse \( {x^2} + 4y^2 = 4 \) meets the tangents at the extremities of its major axis at \( B \) and \( C \), then the circle with \( BC \) as diameter passes through the point:

If the line \( \alpha x + 2y = 1 \), where \( \alpha \in \mathbb{R} \), does not meet the hyperbola \( x^2 - 9y^2 = 9 \), then a possible value of \( \alpha \) is:

The point \( P(-2\sqrt{6}, \sqrt{3}) \) lies on the hyperbola \( \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \) having eccentricity \( \dfrac{\sqrt{5}}{2} \). If the tangent and normal at \( P \) to the hyperbola intersect its conjugate axis at the points \( Q \) and \( R \) respectively, then \( QR \) is equal to:

The length of the latus-rectum of the ellipse, whose foci are $(2, 5)$ and $(2, -3)$ and eccentricity is $\frac{4}{5}$, is:

A triangle is formed by the tangents at the point \( (2, 2) \) on the curves \( y^2 = 2x \) and \( x^2 + y^2 = 4x \), and the line \( x + y + 2 = 0 \). If \( r \) is the radius of its circumcircle, then \( r^2 \) is equal to ________.

If vertex of a parabola is \( (2,-1) \) and the equation of its directrix is \( 4x - 3y = 21 \), then the length of its latus rectum is:

Let the foci and length of the latus rectum of an ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \ a > b \) be \( (\pm 5, 0) \) and \( \sqrt{50} \), respectively. Then, the square of the eccentricity of the hyperbola \( \frac{x^2}{b^2} - \frac{y^2}{a^2 b^2} = 1 \) equals:

The locus of the mid points of the chords of the hyperbola \( x^2 - y^2 = 4 \), which touch the parabola \( y^2 = 8x \), is :

Let the eccentricity of the hyperbola \( \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \) be \( \frac{5}{4} \). If the equation of the normal at the point \( \left(\frac{8}{\sqrt{5}}, \frac{12}{5}\right) \) on the hyperbola is \( 8\sqrt{5}\,x+\beta y=\lambda \), then \( \lambda-\beta \) is equal to __________.

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JEE Main 2023 25th Jan Morning Shift

The distance of the point \( (6,-2\sqrt{2}) \) from the common tangent \( y=mx+c,\, m>0 \), of the curves \( x=2y^{2} \) and \( x=1+y^{2} \) is: