Ellipse — JEE Main PYQs

In the line $\alpha x+4y=\sqrt{7}$, where $\alpha \in \mathbf{R}$, 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 $S$ and $S^{\prime }$ be the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ and $\mathrm{P}(\alpha ,\beta )$ be a point on the ellipse in the first quadrant. If …

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 :

If the points of intersection of the ellipses $x^2+2y^2-6x-12y+23=0$ and $4x^2+2y^2-20x-12y+35=0$ lie on a circle of radius r and centre $(\mathrm{a},\mathrm{b})$, then the value of $\mathrm{ab}+18{\mathrm{r}}^2$ is

Let each of the two ellipses ${\mathrm{E}}_1:\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1,(\mathrm{a}>\mathrm{b})$ and ${\mathrm{E}}_2:\frac{{\mathrm{x}}^2}{{\mathrm{A}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{B}}^2}=1,(\mathrm{A}<\mathrm{B})$ …

Let the length of the latus rectum of an ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1,(\mathrm{a}>\mathrm{b})$, be 30. If its eccentricity is the maximum value of the function $f(t)=-\frac{3}{4}+2t-t^2$, then …

Let $(\mathrm{h},\mathrm{k})$ lie on the circle $\mathrm{C}:{\mathrm{x}}^2+{\mathrm{y}}^2=4$ and the point ( $2\mathrm{h}+1,3\mathrm{k}+2$ ) lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$ is equal to ______.

An ellipse has its center at $(1,-2)$, one focus at $(3,-2)$ and one vertex at $(5,-2)$. Then the length of its latus rectum is :

Let an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a<b$, pass through the point $(4,3)$ and have eccentricity $\frac{\sqrt{5}}{3}$. Then the length of its latus rectum is :

Let $P(3\cos \alpha ,2\sin \alpha ),\alpha \ne 0$, be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1,Q$ 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 …

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 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 $\mathrm{P}(3,\alpha )$ lies on E and O is the origin, then the area of $△\mathrm{POS}$ is equal to :

Let $x=9$ be a directrix of an ellipse $E$, whose centre is at the origin and eccentricity is $\frac{1}{3}$. Let $\mathrm{P}(\alpha ,0),\alpha >0$, be a focus of E and AB be a chord passing through P . Then the locus of the mid point of AB is :

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

Let A be the point $(3,0)$ and circles with variable diameter AB touch the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is e , then $72{\mathrm{e}}^2$ is equal to ______.

Let $\mathrm{E}:\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1,\mathrm{a}>\mathrm{b}$ and $\mathrm{H}:\frac{{\mathrm{x}}^2}{{\mathrm{A}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{B}}^2}=1$.

Let the distance between the foci of E and the foci of H be …

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

If $\alpha x+\beta y=109$ is the equation of the chord of the ellipse $\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1$, whose mid point is $\left(\frac{5}{2},\frac{1}{2}\right)$, then $\alpha +\beta$ is equal to

Let the ellipse, $E_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$ and ${\mathrm{E}}_2:\frac{{\mathrm{x}}^2}{{\mathrm{A}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{B}}^2}=1,\mathrm{A}<\mathrm{B}$ have same eccentricity $\frac{1}{\sqrt{3}}$. Let the product of their lengths of …

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

Let ${\mathrm{E}}_1:\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1$ be an ellipse. Ellipses ${\mathrm{E}}_{\mathrm{i}}$'s are constructed such that their centres and eccentricities are same as that of ${\mathrm{E}}_1$, and the length of minor axis of ${\mathrm{E}}_{\mathrm{i}}$ …

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

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

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:

Let C be the circle of minimum area enclosing the ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\frac{1}{2}$ 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 majo …

If $S$ and $S^{\prime }$ are the foci of the ellipse $\frac{x^2}{18}+\frac{y^2}{9}=1$ and P be a point on the ellipse, then $\min \left(\mathrm{SP}.{\mathrm{S}}^{\prime }\mathrm{P}\right)+\max \left({\mathrm{SP}}^{\prime }{\mathrm{S}}^{\prime }\mathrm{P}\right)$ is equal …

A line passing through the point $\mathrm{P}(\sqrt{5},\sqrt{5})$ intersects the ellipse $\frac{{\mathrm{x}}^2}{36}+\frac{{\mathrm{y}}^2}{25}=1$ at $A$ and $B$ such that $(PA)\cdot (PB)$ is maximum. Then $5(P{A}^2+P{B}^2)$ is equal to:

Let for two distinct values of $p$ the lines $y=x+p$ touch the ellipse $\mathrm{E}:\frac{{\mathrm{x}}^2}{4^2}+\frac{{\mathrm{y}}^2}{3^2}=1$ at the points A and B. Let the line $\mathrm{y}=\mathrm{x}$ intersect E at the points C and $D$. Then the area of the quadrilateral …

The centre of a circle C is at the centre of the ellipse $\mathrm{E}:\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1,\mathrm{a}>\mathrm{b}$. Let C pass through the foci $F_1$ and $F_2$ of $E$ such that the circle $C$ and the ellipse E intersec …

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
$\mathrm{A}=\{(\alpha ,\beta )\in \mathbf{R}\times \mathbf{R}:|\alpha -1|\le 4$ and $|\beta -5|\le 6\}$
and
$B=\left\{(\alpha ,\beta )\in \mathbf{R}\times \mathbf{R}:16(\alpha -2{)}^2+9(\beta -6{)}^2\le 144\right\}$.
Then

Let the length of a latus rectum of an ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ be 10 . If its eccentricity is the minimum value of the function $f(\mathrm{t})={\mathrm{t}}^2+\mathrm{t}+\frac{11}{12}$, …

Let ${\mathrm{e}}_1$ and ${\mathrm{e}}_2$ be the eccentricities of the ellipse $\frac{x^2}{b^2}+\frac{y^2}{25}=1$ and the hyperbola $\frac{x^2}{16}-\frac{y^2}{b^2}=1$, respectively. If $\mathrm{b}<5$ and ${\mathrm{e}}_1{\mathrm{e}}_2=1$, then the eccentri …

Let the ellipse $3x^2+p{y}^2=4$ pass through the centre $C$ of the circle $x^2+y^2-2x-4y-11=0$ of radius r. Let ${\mathrm{f}}_1,{\mathrm{f}}_2$ be the focal distances of the point C on the ellipse. Then $6{\mathrm{f}}_1{\mathrm{f}}_2-\mathrm{r}$ is equal to

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 …

Let $f(x)=x^2+9,g(x)=\frac{x}{x-9}$ and $\mathrm{a}=\mathrm{fog}(10),\mathrm{b}=\mathrm{gof}(3)$. If e and l denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{a}+\frac{y^2}{b}=1$, then $8e^2+l^2$ is equal to.

Let P be a parabola with vertex $(2,3)$ and directrix $2x+y=6$. Let an ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$ of eccentricity $\frac{1}{\sqrt{2}}$ pass through the focus of the parabola P. Then the square of the length of the latus rectum of E, is

If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is:

Let ${\mathrm{e}}_1$ be the eccentricity of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $e_2$ be the eccentricity of the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1,\mathrm{a}>\mathrm{b}$, which passes through the foci of …

The length of the chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid point is $\left(1,\frac{2}{5}\right)$, is equal to :

Let P be a point on the ellipse $\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1$. 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 o …

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 latus rectum is $\sqrt{14}$. Then the square of the eccentricity of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is :

If the points of intersection of two distinct conics $x^2+y^2=4b$ and $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ lie on the curve $y^2=3x^2$, then $3\sqrt{3}$ times the area of the rectangle formed by the intersection points is ______

Let $\mathrm{A}(\alpha ,0)$ and $\mathrm{B}(0,\beta )$ be the points on the line $5\mathrm{x}+7\mathrm{y}=50$. Let the point P divide the line segment AB internally in the ratio $7:3$. Let $3\mathrm{x}-25=0$ be a directrix of the ellipse …

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $\alpha$ and the number of persons who speak only Hindi is $\beta$, then the eccentricity of the ellipse …

Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is reciprocal to that of the hyperbola $2x^2-2y^2=1$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is ____.

Let a circle of radius 4 be concentric to the ellipse $15x^2+19y^2=285$. Then the common tangents are inclined to the minor axis of the ellipse at the angle.

Let the ellipse $\mathrm{E}:{\mathrm{x}}^2+9{\mathrm{y}}^2=9$ intersect the positive x - and y -axes at the points A and B respectively Let the major axis of E be a diameter of the circle C . Let the line passing through A and B meet the circle C at the point P . If the area …

Let $P\left(\frac{2\sqrt{3}}{\sqrt{7}},\frac{6}{\sqrt{7}}\right),Q,R$ and $S$ be four points on the ellipse $9x^2+4y^2=36$. Let $PQ$ and $RS$ be mutually perpendicular and pass through the origin. If $\frac{1}{(PQ{)}^2}+\frac{1}{(RS{)}^2}=\frac{p}{q}$, where …

Let the tangent and normal at the point $(3\sqrt{3},1)$ on the ellipse $\frac{x^2}{36}+\frac{y^2}{4}=1$ meet the $y$-axis at the points A and B respectively. Let the circle C be drawn taking AB as a diameter and the line $\mathrm{x}=2\sqrt{5}$ intersect C at the po …