Conic Sections — JEE Main PYQs

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 the parabola $y = x^2 + px + q$ passing through the point $(1, -1)$ be such that the distance between its vertex and the $x$-axis is minimum. Then the value of $p^2 + q^2$ is :

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 __________.

Consider the parabola $P : y^2 = 4kx$ and the ellipse $E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. 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 P $(3\cos\alpha, 2\sin\alpha)$, $\alpha \neq 0$, be a point on the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$, Q be a point on the circle $x^2 + y^2 - 14x - 14y + 82 = 0$ and R be a point on the line $x + y = 5$ such that the centroid of the triangle PQR is $\left(2 + \cos\alpha,\ 3 + \dfrac{2}{3}\sin\alpha\right)$. Then the sum of the ordinates of all possible points R is:

Let $H : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is $\dfrac{8}{3}$. 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 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 $\triangle ABC$ be a conic $C_o$. Then three times the length of latus rectum of $C_o$ is __________

Let a focus of the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be $S(4, 0)$ and its eccentricity be $\frac{4}{5}$. If the point $P(3, \alpha)$ lies on $E$ and $O$ is the origin, then the area of $\triangle POS$ 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 $3/5$. If $BC$ is a focal chord of $P$, then six times the area of $\Delta ABC$ is :

Let the eccentricity $e$ 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 $e_1$ and $e_2$ be two distinct roots of the equation $x^2-ax+2=0$. Let the sets

$\{a \in \mathbb{R} : e_1 \text{ and } e_2 \text{ are the eccentricities of hyperbolas}\} = (\alpha, \beta)$, and

$\{a \in \mathbb{R} : e_1 \text{ and } e_2 \text{ are the eccentricities of an ellipse and a hyperbola, respectively}\} = (\gamma, \infty)$.

Then $\alpha^2+\beta^2+\gamma^2$ is equal to:

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

The eccentricity of an ellipse E with centre at the origin O is $\frac{\sqrt{3}}{2}$ and its directrices are $x = \pm \frac{4\sqrt{6}}{3}$. Let $H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ 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 $x = 9$ be a directrix of an ellipse E, whose centre is at the origin and eccentricity is $\frac{1}{3}$. Let $P(\alpha, 0)$, $\alpha > 0$, be a focus of E and AB be a chord passing through P. Then the locus of the mid point of AB is :

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 $\frac{x^2}{f\left(a^2 + 7a + 3\right)} + \frac{y^2}{f(3a + 15)} = 1$ represent an ellipse with major axis along $y$-axis, where $f$ is a strictly decreasing positive function on $\mathbb{R}$. If the set of all possible values of $a$ is $\mathbb{R} - [\alpha, \beta]$, then $\alpha^2 + \beta^2$ is equal to :

Let the focal chord PQ of the parabola $y^2 = 4x$ make an angle of $60°$ 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 :

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 $\min(\mathrm{SP} \cdot \mathrm{S'P}) + \max(\mathrm{SP} \cdot \mathrm{S'P})$ is equal to :

Let one focus of the hyperbola $\mathrm{H}: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be at $\left(\sqrt{10}, 0\right)$ 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 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 length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :

The radius of the smallest circle which touches the parabolas $y = x^2 + 2$ and $x = y^2 + 2$ is

A line passing through the point $P\left(\sqrt{5}, \sqrt{5}\right)$ 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 product of the focal distances of the point $\mathrm{P}\left(4, 2\sqrt{3}\right)$ on the hyperbola $\mathrm{H}: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ 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 $C$ 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 :

The shortest distance between the curves $y^2 = 8x$ and $x^2 + y^2 + 12y + 35 = 0$ 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 __________.

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

Let $C$ be the circle $x^2 + (y-1)^2 = 2$, $E_1$ and $E_2$ be two ellipses whose centres lie at the origin and major axes lie on x-axis and y-axis respectively. Let the straight line $x + y = 3$ touch the curves $C$, $E_1$ and $E_2$ at $P(x_1, y_1)$, $Q(x_2, y_2)$ and $R(x_3, y_3)$ respectively. Given that $P$ is the mid point of the line segment $QR$ and $PQ = \frac{2\sqrt{2}}{3}$, the value of $9(x_1 y_1 + x_2 y_2 + x_3 y_3)$ 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 $(1,\mathrm{k})$ lies on the parabola, then a possible value of $\mathrm{k}$ is :

Let for two distinct values of $p$ the lines $y=x+p$ touch the ellipse $\mathrm{E}:\dfrac{x^2}{4^2}+\dfrac{y^2}{3^2}=1$ 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 centre of a circle C is at the centre of the ellipse $\mathrm{E}:\dfrac{x^2}{\mathrm{a}^2}+\dfrac{y^2}{\mathrm{b}^2}=1$, $\mathrm{a}>\mathrm{b}$. Let C pass through the foci $\mathrm{F}_1$ and $\mathrm{F}_2$ 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 $\mathrm{PF}_1\mathrm{F}_2$ 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 distances of the point $\mathrm{P}(4,3)$ on the hyperbola $\mathrm{H}:\dfrac{x^2}{\mathrm{a}^2}-\dfrac{y^2}{\mathrm{b}^2}=1$ 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 $\mathrm{m}$, then $9l^2+6\mathrm{m}$ is equal to :

Let P be the parabola, whose focus is $(-2, 1)$ and directrix is $2x + y + 2 = 0$. Then the sum of the ordinates of the points on P, whose abscissa is $-2$, is

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

Let the length of a latus rectum of an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ be 10. If its eccentricity is the minimum value of the function $f(t) = t^2 + t + \dfrac{11}{12}$, $t \in \mathbf{R}$, then $a^2 + b^2$ is equal to :

Let $e_1$ and $e_2$ be the eccentricities of the ellipse $\dfrac{x^2}{b^2} + \dfrac{y^2}{25} = 1$ and the hyperbola $\dfrac{x^2}{16} - \dfrac{y^2}{b^2} = 1$, respectively. If $b < 5$ and $e_1 e_2 = 1$, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is :

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be $2a$ and $2b$, respectively, and one focus and the corresponding directrix of this hyperbola be $(-5, 0)$ and $5x + 9 = 0$, respectively. If the product of the focal distances of a point $\left(\alpha, 2\sqrt{5}\right)$ on the hyperbola is $p$, then $4p$ is equal to _________.

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 $6f_1 f_2 - r$ 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 :

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 :

Let $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $a > b$ and $H : \dfrac{x^2}{A^2} - \dfrac{y^2}{B^2} = 1$. 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 $\dfrac{1}{3}$, then the sum of the lengths of their latus rectums 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

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

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, and having its centre on the axis of the parabola is :

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

If the equation of the parabola with vertex $\mathrm{V}\left(\dfrac{3}{2}, 3\right)$ and the directrix $x + 2y = 0$ is $\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0$, then $\alpha + \beta + \gamma$ is equal to :

The equation of the chord, of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$, whose mid-point is $(3, 1)$ is :

Let $\mathrm{H_1} : \dfrac{x^2}{\mathrm{a}^2} - \dfrac{y^2}{\mathrm{b}^2} = 1$ and $\mathrm{H_2} : -\dfrac{x^2}{\mathrm{A}^2} + \dfrac{y^2}{\mathrm{B}^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their ecentricities be $\mathrm{e_1} = \sqrt{\dfrac{5}{2}}$ and $\mathrm{e_2}$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25\mathrm{e_2^2}$ 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

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 $E_{i+1}$ $(i \ge 1)$. If $A_i$ is the area of the ellipse $E_i$, then $\frac{5}{\pi}\left(\sum_{i=1}^{\infty} A_i\right)$, is equal to ____.

If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $\left(\sqrt{2},\ \frac{4}{3}\right)$, and the length of the chord is $\frac{2\sqrt{\alpha}}{3}$, then $\alpha$ is :

If A and B are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ and a point P moves on the line $2x - 3y + 4 = 0$, then the centroid of $\Delta \mathrm{PAB}$ lies on the line :

Let A and B be the points of intersection of the mirror image of the parabola $y^2 = 4x$ in the line $x + y + 4 = 0$, and the line $y + 5 = 0$. If the distance between A and B is $d$ and the area of $\Delta \mathrm{SAB}$ is $a$, where S is the focus of the parabola $y^2 = 4x$, then the value of $(a + d)$ is __________. [English block corrupted in the source PDF; transcribed from the Hindi duplicate of the same question]

Let A be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is $-3$. If the points $(x, y)$ satisfying $A^{2}+xA+y\,I=O$ lie on a hyperbola, whose transverse axis is parallel to the $x$-axis, eccentricity is $e$ and the length of the latus rectum is $l$, then $e^{4}+l^{4}$ 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 ________.

Consider a hyperbola H having centre 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 units) 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 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?

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 $\dfrac{\mathrm{m}}{\mathrm{n}}$, where m and n are coprime, then $2\mathrm{m} + \mathrm{n}$ is equal to

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

Let a line perpendicular to the line $2x - y = 10$ touch the parabola $y^2 = 4(x - 9)$ at the point P. The distance of the point P from the centre of the circle $x^2 + y^2 - 14x - 8y + 56 = 0$ is ________.

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 a conic $C$ pass through the point $(4, -2)$ and $P(x, y)$, $x \geq 3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $(3, -5)$. If the focal distance of the point $(7, 1)$ on $C$ is $d$, then $12d$ equals ________.

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 $4\mathrm{e}^2+\mathrm{m}$ 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 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 $\mathrm{A}\left(\sqrt{6}, \sqrt{5}\right)$ 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 $f(x)=x^2+9$, $g(x)=\frac{x}{x-9}$ and $\mathrm{a}=f{\circ}g(10)$, $\mathrm{b}=g{\circ}f(3)$. If e and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{\mathrm{a}}+\frac{y^2}{\mathrm{b}}=1$, then $8\mathrm{e}^2+l^2$ is equal to.

Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E: \dfrac{(x-1)^2}{100}+\dfrac{(y-1)^2}{75}=1$ 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 $A$, $B$ 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 the perpendiculars from $A$ and $B$ on $L$. Then $\left(\dfrac{AM \cdot BN}{CD}\right)^2$ is equal to ________.

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

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