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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:

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 \( E_1 : \frac{x^2}{9} + \frac{y^2}{4} = 1 \) 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 ______.

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 __________.

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 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 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:

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 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 \( C \) 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:

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:

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:

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 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:

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:

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 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:

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

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

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

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 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:

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:

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 ________.

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

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

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JEE MAIN 2025 4th April Morning Shift

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