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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:

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:

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:

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 \( A \) 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 \( O \) 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:

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:

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:

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 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:

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:

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 \( e_{1} \) 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:

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:

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 ________.

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 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:

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 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 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 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:

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 \( (h,k) \) 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 ____________.

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:

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:

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 ________.

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:

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:

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JEE Main 2026 24th Jan Evening Shift

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 ____________.