Ellipse Questions - JEE

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 area of the largest triangle that an incident ray and corresponding reflected ray can enclose with the major axis of the ellipse is equal to:

Consider an ellipse $\frac{x^2}{36} + \frac{y^2}{18} = 1$. There is a hyperbola whose one asymptote is the major axis of the given ellipse. If eccentricity of the given ellipse and hyperbola are reciprocal to each other, both have the same centre and both touch each other in the first and third quadrants. Find the equation of the common tangent to the given ellipse and hyperbola in the first quadrant.

The point of intersection of the tangents at the point P on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and its corresponding point Q on the auxiliary circle meet on the line:

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

If $3x + 4y = 12\sqrt{2}$ is a tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{9} = 1$ for some $a \in \mathbb{R}$, then the distance between the foci of the ellipse is

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:

It is given that eccentricity is $e = \dfrac{3}{5}$ and the distance between the foci is $2ae = 6$. Find the area of the quadrilateral formed by the axes of the ellipse.

The equation of the pair of tangents drawn from the point (1, 2) to the ellipse $3x^2 + 2y^2 = 5$ is

Let $\mathrm{P}(\mathrm{p} \sec \theta, \mathrm{q} \tan \theta)$ and $\mathrm{Q}(\mathrm{p} \sec \phi, \mathrm{q} \tan \phi)$ where $\theta+\phi=\frac{\pi}{2}$, be two points on the hyperbola $\frac{x^2}{p^2}-\frac{y^2}{q^2}=1$. If $\left(x_1, y_1\right)$ is the point of intersection of normals at $P$ and $Q$, then $\mathrm{y}_1$ is equal to

Tangents are drawn to the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b)$ and the circle $x^2 + y^2 = a^2$ at the points where a common ordinate cuts them (on the same side of the $x$-axis). Then the greatest acute angle between these tangents is given by

Consider an ellipse $\frac{x^2}{36} + \frac{y^2}{18} = 1$. There is a hyperbola whose one asymptote is the major axis of the given ellipse. If eccentricity of the given ellipse and hyperbola are reciprocal to each other, both have the same centre and both touch each other in the first and third quadrants. Find the focus of the hyperbola.

An ellipse has OB as a semi-minor axis, F, F' as its foci and the angle $\angle$FBF' is a right angle. Then, the eccentricity of the ellipse is

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 eccentricity of the ellipse is equal to:

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 the eccentricity of an ellipse be $\frac 58$ and the distance between its foci be 10, then its latus rectum is:

The equation $14x^2 - 4xy + 11y^2 - 44x - 58y + 71 = 0$ whose centre is

The equation of the ellipse whose centre is at origin and which passes through the points (-3, 1) and (2, -2) is

If the normal at one end of latus rectum of ellipse $ \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 $ passes from one end of minor axis and e is eccentricity of ellipse, then:

The foci of the ellipse 25x2 + 50x + 9y2 + 36y - 164 = 0 are

The normal at one of the ends of a latus rectum of the ellipse b2x2 + a2y2 = a2b2 passes through the end of the minor axis. Then e satisfies

A vertical line passing through the point $(h, 0)$ intersects the ellipse $\frac{x^2}{4} + \frac{y^2}{3} = 1$ at the points P and Q. Let the tangents to the ellipse at P and Q meet at the point R. If $D(h)$ = area of the triangle PQR, $D_1 = \max_{1/2 \leq h \leq 1} D(h)$ and $D_2 = \max_{1/2 \leq h \leq 1} D(h)$, then $D_1 - 8D_2 = $ ?

The equation of the locus of the point whose distances from the point P and the line AB are equal, is:

The line $x = 8$ is the directrix of the ellipse $E: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ with the corresponding focus $(2, 0)$. If the tangent to $E$ at the point $P$ in the first quadrant passes through the point $\left(0, 4\sqrt{3}\right)$ and intersects the x-axis at Q, then $(3PQ)^{2}$ is equal to ____.

The equations of tangents to the ellipse $4x^2 + 3y^2 = 5$, which are inclined at an angle of 60° to the X-axis is

If B is one end of the minor axis of an ellipse and F1, F2 are its foci such that ∠F1BF2 = 90°, then the eccentricity of the ellipse is

Statement-1: An equation of a common tangent to the parabola $y^2 = 16\sqrt{3}x$ and the ellipse $2x^2 + y^2 = 4$ is $y = 2x + 2\sqrt{3}$.Statement-2: If the line $y = mx + \dfrac{4\sqrt{3}}{m}$, $(m \neq 0)$ is a common tangent to the parabola $y^2 = 16\sqrt{3}x$ and the ellipse $2x^2 + y^2 = 4$, then $m$ satisfies $m^4 + 2m^2 = 24$.

An ellipse slides between two lines at right angles to one another. Then, the locus of its centre 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:

264. Given that $m, n, s, t \in (0, +\infty)$, $m + n = 3$, $\dfrac{m}{s} + \dfrac{n}{t} = 1$, $m, n$ are constants and \(m

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 $(a^2+b^2)$ is equal to

The equations of tangent and normal at point (3, -2) of ellipse 4x2 + 9y2 = 36 are

Let $d$ be the perpendicular distance from the centre of the ellipse to any tangent to the ellipse. If $F_1$ and $F_2$ are the two foci of the ellipse, then

The eccentricity of an ellipse whose centre is at the origin is $1/2$. If one of its directrices is $x = -4$, then the equation of the normal to it at $\left(1, \dfrac{3}{2}\right)$ is

Let $O(0, 0)$ and $A(0, 1)$ be two fixed points. Then the locus of a point $P$ such that the perimeter of $\triangle AOP$ is 4, is:

An ellipse passes through the foci of the hyperbola, $9x^2 - 4y^2 = 36$ and its major and minor axes lie along the transverse and conjugate axes of the hyperbola, respectively. If the product of eccentricities of the two conics is $1/2$, then which of the following points does not lie on the ellipse?

A tangent to the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{81} = 1$ at the point $(b\cos\phi,\, a\sin\phi)$ meets the coordinate axes. Find the maximum area of the triangle formed by the tangent and the coordinate axes.

The orthocentre of the triangle PAB is:

In right angled triangle FBF', F and F' are foci of an ellipse. If the angle at B is a right angle, find the eccentricity of the ellipse.

Question 648Consider, $E : \dfrac{(x-1)^2}{16} + \dfrac{(y-2)^2}{9} = 1$ 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 incorrect combination?

Let $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (a > b) be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function $f(t) = \frac{5}{12} + t - t^2$, then $a^2 + b^2$ is equal to

If the value of $\sum_{i=1}^{n} \dfrac{\text{Area}(\Delta P_i T_i S) \cdot \text{Area}(\Delta P_i T_i S')}{(P_i T_i)^2} = 18$, where $S$ and $S'$ represents the foci of the ellipse, then $n$ equal to:

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 foci of the ellipse is:

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

Ellipse Questions (292)