Ellipse Questions - JEE

If $OB$ is the semi-minor axis of an ellipse, $F_1$ and $F_2$ are its foci and the angle between $F_1B$ and $F_2B$ is a right angle, then the square of the eccentricity of the ellipse is

The angle between the tangents drawn from the point $(7, 1)$ to the ellipse $3x^2 + 5y^2 = 15$ is:

If , then the chord joining two points and on the ellipse will subtend a right angle at

Consider a family of ellipses \ $\frac{x^2}{a_n^2} + \frac{y^2}{b_n^2} = 1$ $(a_n > b_n)$, where $b_n^2 = a_n^2(1 - e_n^2)$. For $E_{n-1}$, $a_{n-1}^2 = b_{n-1}^2(1 - e_{n-1}^2)$, $b_n = b_{n-1}e_{n-1}$, $a_{n-1} = a_n$. Let all the eccentricities be $e$. Find the value of $e$.

Let an ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$ and a parabola $x^2 = 2y$ intersecting each other above x-axis at P and Q. Tangents to ellipse at P and Q intersect y-axis at R and tangents to parabola at P and Q intersect y-axis at S, then the area of quadrilateral PRQS is $\lambda\sqrt{3}$, then $[\lambda]$ = (where [.] denote greatest integer function)

A coplanar beam of light emerging from a point source have the equation $lx - y + 2(1 + l) = 0$, $l \in \mathbb{R}$; the rays of the beam strike an elliptical surface and get reflected inside the ellipse. The reflected rays form another convergent beam having the equation $mx - y + 2(1 - m) = 0$, $m \in \mathbb{R}$. Further it is found that the foot of the perpendicular from the point (2, 2) upon any tangent to the ellipse lies on the circle $x^2 + y^2 - 4y - 5 = 0$. The least value of total distance travelled by an incident ray and the corresponding reflected ray is equal to:

Question 647Consider, $E : \dfrac{(x-1)^2}{16} + \dfrac{(y-2)^2}{9} = 1$ and $H : (x-1)^2 - (y-2)^2 = \dfrac{7}{2}$.(Refer to the match the column table for questions 644–648.)Which of the following options is the only correct combination?

A point on the ellipse $x^2 + 3y^2 = 37$ where the normal is parallel to the line $6x - 5y = 2$, is

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

Ellipse Questions (292)