Ellipse Questions - JEE

Let an ellipse with centre $(1,0)$ and latus rectum of length $\dfrac{1}{2}$ have its major axis along the $x$-axis. If its minor axis subtends an angle $60°$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to ___.

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 the line $x - 2y = 12$ is tangent to the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ at the point $\left(3, \dfrac{-9}{2}\right)$, then the length of the latus rectum of the ellipse is

Consider the particle travelling clockwise on the elliptical path $\frac{x^2}{100} + \frac{y^2}{25} = 1$. The particle leaves the orbit at the point (-8, 3) and travels in a straight line tangent to the ellipse. At what point will the particle cross the y-axis?

Let the ellipse $E_1 : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $a > b$ and $E_2 : \dfrac{x^2}{A^2} + \dfrac{y^2}{B^2} = 1$, $A < B$ have the same eccentricity $\dfrac{1}{\sqrt{3}}$. Let the product of their lengths of latus rectums be $\dfrac{32}{\sqrt{3}}$, and the distance between the foci of $E_1$ be $4$. If $E_1$ and $E_2$ meet at $A$, $B$, $C$ and $D$, then the area of the quadrilateral $ABCD$ equals

If a tangent of slope 2 of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ passes through the point $(-2, 0)$, then the value of $a^2$ is equal to

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

The eccentricity of an ellipse, with its centre at the origin, is $1/2$. If one of the directrices is $x = 4$, then the equation of the ellipse is

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

If the tangent at the point $\left(4\cos\theta, \frac{16\sin\theta}{11}\right)$ to the ellipse $16x^2 + 11y^2 = 256$ is also tangent to the circle $x^2 + y^2 - 2x = 15$, then $\theta$ equals:

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

The number of normals that can be drawn to an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ from an exterior point is ___ in general.

An ellipse having foci at (3, 3) and (-4, 4) and passing through the origin has eccentricity equal to:

The point, which is at the shortest distance from the line $x + y = 7$ and lying on an ellipse $x^2 + 2y^2 = 6$, has coordinates $(a, b)$ then the value of $\frac{a}{b}$ is

If $\alpha x + \beta y = 109$ is the equation of the chord of the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$, whose mid point is $\left(\dfrac{5}{2}, \dfrac{1}{2}\right)$, then $\alpha + \beta$ 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 $\dfrac{m}{n}$, where $m$ and $n$ are coprime, then $2m+n$ is equal to:

The eccentricity of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$ = 1 whose latus rectum is half of its major axis is:

The length of sides of square which can be made by four perpendicular tangents to the ellipse $\frac{x^2}{7} + \frac{2y^2}{11} = 1$ is:

For two confocal conics with b' = a, a'e' = b, and 4r^2 = 4a^2 + 4b^2, find the eccentricity.

If normal at any point P to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b$) meet the axes at M and N so that $\frac{PM^2}{PN^2} = \frac{2}{3}$, then the value of eccentricity 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:

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

If $P$ and $Q$ are points with eccentric angles $\theta$ and $\left(\theta + \frac{\pi}{2}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, then the area (in sq. units) of the triangle $OPQ$ (where $O$ is the origin) is equal to

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:

If the line joining foci subtends an angle of $90°$ at an extremity of minor axis then the eccentricity of the ellipse is:

The length of the chord of the ellipse 2 y , whose mid-point is (1, , is : x 1 + = 1 ) 4 2 2

An ellipse passes through the points $(4, -1)$ and $(-2, 2)$. Find the required eccentricity of the ellipse.

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

A circle concentric to the ellipse $\frac{4x^2}{289} + \frac{y^2}{l} = 1$ \(\left(l

Let the ellipse $E:\dfrac{x^2}{144}+\dfrac{y^2}{169}=1$ and the hyperbola $H:\dfrac{y^2}{z^2}-\dfrac{x^2}{\lambda^2}=-1$ have the same foci. If $e$ and $L$ respectively denote the eccentricity and the length of the latus rectum of $H$, then the value of $24(e+L)$ is:

The eccentricity of the curve represented by the equation x2 + 2y2 - 2x + 3y + 2 = 0 is

If straight line $\frac{ax}{3} + \frac{by}{4} = c$ is a normal to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b$, then $a^2 - b^2$ is equal to:

If $S$ and $S'$ are the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ and $P$ is any point on it, then the difference of maximum and minimum of $SP \cdot S'P$ is equal to:

Rectangle $ABCD$ has area $200$. An ellipse with area $200\pi$ passes through $A$ and $C$ and has foci at $B$ and $D$. Let perimeter of the rectangle $ABCD$ is $P$, then $\frac{P}{10} =$

The line $3x + 5y = k$ touches the ellipse $16x^2 + 25y^2 = 400$ if $k$ is

An ellipse is inscribed in a circle and a point within the circle is chosen at random. If the probability that this point lies outside the ellipse is \frac{2}{3} then the eccentricity of the ellipse is:

If the tangents on the ellipse $4x^2 + y^2 = 8$ at the points $(1, 2)$ and $(a, b)$ are perpendicular to each other, then $a^2$ is equal to __________ (up to four decimal places).

If the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{b^2}=1$ and the hyperbola $\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}$ coincide, then the value of $b^2$ is.

The tangent and normal to the ellipse $3x^2 + 5y^2 = 32$ at the point P(2, 2) meet the $x$-axis at Q and R, respectively. Then the area (in sq. units) of the triangle PQR is __________ (up to three decimal places).

Equation of foci of an ellipse are $(\pm 2, 0)$ and $e = \frac{1}{2}$. Then the equation of the ellipse is:

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

Let the equation of ellipse be $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ where \(a

A stair-case of length $l$ rests against a vertical wall and a floor of a room. Let $P$ be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio $1:2$. If the stair-case begins to slide on the floor, then the locus of $P$ is

For the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{5} = 1$, one end of the latus rectum is at $P(ae, b^2/a)$. Find the equation of the tangent at this point.

If the foci of an ellipse are (0, $\pm$4) and the equation of the directrices are y = $\pm$9, then the equation of the ellipse is

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

The eccentricity of an ellipse having centre at the origin, axes along the coordinate axes and passing through the points $(4, -1)$ and $(-2, 2)$ is

Let the length of the latus rectum of an ellipse with its major axis along X-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?

(A) The minimum area of triangle formed by the tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and coordinate axes is:

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

Ellipse Questions (292)