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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:

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;

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:

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.

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 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 \( \_\_\_\_\_\_\_\_ \).

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 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 ______.

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:

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 \( \_\_\_\_\_\_\_\_\_\_ \).

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 \( PQ \) 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 \( Q \)?

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 ________.

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 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 ________.

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:

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 \( P \) 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:

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 \( A(\alpha, 0) \) 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:

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 \( H : \frac{-x^2}{a^2} + \frac{y^2}{b^2} = 1 \) 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

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 \( e_1 \) 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:

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:

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JEE Main 2024 30th Jan Evening Shift

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.