Ellipse Questions - JEE

The ellipse $E_1: \frac{x^2}{9} + \frac{y^2}{4} = 1$ is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse $E_2$ passing through the point $(0, 4)$ circumscribes the rectangle R. The eccentricity of the ellipse $E_2$ is:

A normal is drawn to the ellipse $\frac{x^2}{(a^2+2a+2)^2} + \frac{y^2}{(a^2+1)^2} = 1$ whose centre is at O. If maximum radius of the circle, centered at the origin and touching the normal, is 5 then find the positive value of $a$.

A circle has the same centre as an ellipse & passes through the foci F_1 & F_2 of the ellipse, such that the two curves intersect in 4 points. Let 'P' be any one of their point of intersection. If the major axis of the ellipse is 17 & the area of the triangle PF_1F_2 is 30, then the distance between the foci is:

An ellipse and hyperbola intersect orthogonally. They must be confocal. Let point $P(2, 3)$ lies on both the curves. If the equations are $4a^2x^2 + 9b^2y^2 = 36$ ...(i) and $4a^2x^2 - 3b^2y^2 = 4$ ...(ii), find $a^2 + b^2$.

An ellipse has semi-major axis of length 2 and semi-minor axis of length 1. It slides between the co-ordinate axes in the first quadrant, while maintaining contact with both x-axis and y-axis. The locus of the centre of ellipse is:

Variable pairs of chords at right angles are drawn through a point $ P $ (with eccentric angle $ \dfrac{x}{4} $) on the ellipse $ \dfrac{x^2}{4} + y^2 = 1 $ to meet the ellipse at two points, say $ A $ and $ B $. If the line joining $ A $ and $ B $ passes through a fixed point $ Q = (a, b) $ and the value of $ a^2 + b^2 $ can be expressed as $ \dfrac{m}{n} $, where $ m $ and $ n $ are co-prime positive integers, submit your answer as $ n - m $.

If normal at $P\left(2, \frac{3 \sqrt{3}}{2}\right)$ meet the major axis of ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ at $Q$ and $S^{\prime}$ and $S$ are foci of given ellipse then SQ: $S^{\prime} Q$ is

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

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:

The angle between the pair of tangents drawn to the ellipse 3x2 + 2y2 = 5 from the point (1, 2) is

Let S and S' be the foci of the ellipse and B be any one of the extremities of its minor axis. If $\Delta S'BS$ is a right angled triangle with right angle at B and area ($\Delta S'BS$) = 8 sq. units, then the length of a latus rectum of the ellipse is :

Let x = 4 be a directrix to an ellipse whose centre is at origin and its eccentricity is e = 1/2. If P(1, b), b > 0 is a point on this ellipse, then the equation of the normal to it at P is

Let C be the largest circle centred at $(2, 0)$ and inscribed in the ellipse $\frac{x^{2}}{36} + \frac{y^{2}}{16} = 1$. If $(1, \alpha)$ lies on C, then $10\alpha^{2}$ is equal to ____.

Let $P$ be a point on the ellipse $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$. 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:

In the above problem, if $m = \frac{\text{area of } \triangle PQR}{\text{area of } \triangle PQS}$, then $2m$ =

An ellipse, with foci at (0, 2) and (0, -2) and minor axis of length 4, passes through which of the following points?

If e1 and e2 are the eccentricities of the ellipse, $\frac{x^2}{18} + \frac{y^2}{4} = 1$ and the hyperbola, $\frac{x^2}{9} - \frac{y^2}{4} = 1$ respectively and (e1, e2) is a point on the ellipse, $15x^2 + 3y^2 = k$, then k is equal to

The length of the minor axis (along Y-axis) of an ellipse in the standard form is $\frac{4}{3}$. If this ellipse touches the line $x + 6y = 8$, then find its eccentricity.

Let P(1, 2) be a point from which pair of tangents are drawn to the ellipse $\frac{x^{2}}{4}$ + y2 = 1. Tangents touch the ellipse at points A and B. If the point of intersection of normals at points A and B is Q, then equation of PQ is:

For the ellipse $\frac{x^2}{36} + \frac{y^2}{18} = 1$ and the hyperbola described above, find the equation of the common tangent in the first quadrant.

The locus of the foot of perpendicular drawn from the centre of the ellipse $x^2 + 3y^2 = 6$ on any tangent to it is

If the midpoint of a chord of the ellipse 2 y 2\sqrt\alpha is (\sqrt2, 4/3), and the length of the chord is , then \alpha is : x + = 1 9 4 3

The equation of ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Parametric equation is $x = a\cos\theta,\ y = b\sin\theta$. Let the coordinates of $A$ be $(a\cos\theta,\ b\sin\theta)$; $\theta \in (0, \pi/2)$. A rectangle is inscribed in the ellipse with sides parallel to the axes. The maximum area of the rectangle inscribed in the ellipse is:

Consider ellipses $E_k:\ kx^2+k^2y^2=1$, $k=1,2,\ldots,20$. Let $C_k$ be the circle which touches the four chords joining the end points (one on minor axis and one on major axis) of $E_k$. If $r_k$ is the radius of $C_k$, then $\displaystyle\sum_{k=1}^{20}\dfrac{1}{r_k^2}$ is equal to

C is the centre of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ and A and B are two points on the ellipse such that $\angle ACD = 90°$. Then $\frac{1}{CA^2} + \frac{1}{CB^2} = $:

Given $x^2 + 3y^2 = 9$, i.e., $\dfrac{x^2}{9} + \dfrac{y^2}{3} = 1$. The equation of tangent at point $(3\cos\theta, \sqrt{3}\sin\theta)$ is $\dfrac{x\cos\theta}{3} + \dfrac{y\sin\theta}{\sqrt{3}} = 1$. The equation of tangent at point $(-3\sin\theta, \sqrt{3}\cos\theta)$ is $\dfrac{-x\sin\theta}{3} + \dfrac{y\cos\theta}{\sqrt{3}} = 1$. For the two tangents to be perpendicular, which of the following is correct?

The length of the latus rectum of the ellipse 5x2 + 9y2 = 45 is

If the area of the quadrilateral formed by the tangents at the ends of the latus rectum of the ellipse $E$ is $\dfrac{16\lambda}{\sqrt{55}}$, then $\lambda$ equals:

If the line $x\cos\alpha + y\sin\alpha = p$ be normal to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, then

The orthocentre of the triangle PAB is:

The eccentricity of the ellipse which meets the straight line $\frac{x}{7} + \frac{y}{2} = 1$ on the axis of x and the straight line $\frac{x}{3} - \frac{y}{5} = 1$ on the axis of y and whose axis lie along the axes of coordinates is:

Let an ellipse having major axis and minor axis parallel to x-axis and y-axis respectively. Its two foci S and S' are (2, 1), (4, 1) and a line x + y = 9 is a tangent to this ellipse at point P. Find the length of major axis.

If the tangent at a point on the ellipse $\dfrac{x^2}{27} + \dfrac{y^2}{3} = 1$ meets the coordinates axes at $A$ and $B$, and $O$ is the origin, then the minimum area (in sq. units) of the triangle $OAB$ is

The point on the ellipse $x^2 + 2y^2 = 6$ closest to the line $x + y = 7$:

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

If the line $2px + y\sqrt{1-p^2} = 1$ always touches the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $\forall\, p \in (-1,1) - \{0\}$. The eccentricity of this ellipse, is

The equation of the normal at the point (2, 3) on the ellipse $9x^2 + 16y^2 = 180$ is

Which one of the following is the common tangent to the ellipses $\frac{x^2}{a^2 + b^2} + \frac{y^2}{b^2} = 1$ and $\frac{x^2}{a^2} + \frac{y^2}{a^2 + b^2} = 1$?

The equation of the ellipse whose foci are $(\pm 2, 0)$ and eccentricity is $1/2$, is

Let E : 2 2 y be an ellipse. Ellipses E 's are constructed such that their centres and eccentricities are x 1 + = 1 1 9 4 same as that of E , and the length of minor axis of E is the length of major axis of E 1 i i+1 (i \ge 1) . If A is the area i of the ellipse E , then , is equal to 5 \infty i (\sum Ai ) \pi i=1

If the latus rectum of an ellipse be equal to half of its minor axis, then its eccentricity is:

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 ellipse E : xa 2 + yb 2 = 1, a > b and E 2 : xA 2 + yB 2 = 1, A < B have same eccentricity 1 1 32 . Let the product of their lengths of latus rectums be , and the distance between the foci of E 1 be 4. If 3 3 E 1 and E 2 meet at A, B, C and D , then the area of the quadrilateral ABCD equals : 6

Let $E_1 : \dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ 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 $\dfrac{5}{\pi}\left(\sum_{i=1}^{\infty} A_i\right)$ is equal to _____.

If \alphax + \betay = 109 is the equation of the chord of the ellipse 2 y , whose mid point is ( , then \alpha + \beta x 5 1 + = 1 , ) 9 4 2 2 is equal to :

The area of the rectangle formed by the perpendiculars from the centre of the standard ellipse to the tangent and normal at its point whose eccentric angle is $\frac{\pi}{4}$, is:

The equation of the circle passing through the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$, and having centre at $(0, 3)$ is:

An ellipse is drawn by taking a diameter of the circle $(x-1)^2 + y^2 = 1$ as its semi-minor axis and a diameter of the circle $x^2 + (y-2)^2 = 4$ as its semi-major axis. If the centre of the ellipse is the origin and its axes are the coordinate axes, then the equation of the ellipse is

The locus of point of intersection of perpendicular tangents of ellipse $\frac{(x-1)^2}{16} + \frac{(y-1)^2}{9} = 1$ is:

The semi-minor axis is $b = 2$ and the semi-major axis is $a = 4$. Find the equation of the ellipse.

Ellipse Questions (292)