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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 6th April Evening Shift

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: