Ellipse Questions - JEE

If the variable line $y=kx+2h$ is a tangent to the ellipse $2x^2+3y^2=6$, then the locus of $P(h,\frac{5k}{2})$ is a conic $C$ whose eccentricity equals:

Consider the ellipse $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$. Let $S(p,q)$ be in the first quadrant with $\dfrac{p^2}{9}+\dfrac{q^2}{4}>1$. Two tangents from $S$: one at a minor axis endpoint, the other at $T$ in the fourth quadrant. $R$ = vertex with positive $x$-coordinate, $O$ = center. If area of $\triangle ORT=\dfrac{3}{2}$, then:

Let $a$, $b$, $k$ be positive real numbers. Suppose $Q$ is a point on the parabola $y^2=4kx$ such that its distance from the focus is $2k$ and it lies in the first quadrant. An ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ passes through $Q$. If the normal to the parabola at $Q$ and the tangent to the ellipse at $Q$ are parallel, then the eccentricity of the ellipse is:

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 $\min \left({SP} \cdot {S}^{\prime} {P}\right)+\max \left({SP} \cdot {S}^{\prime} {P}\right)$ is equal to:

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

Let the ellipse $3 x^{2}+p y^{2}=4$ pass through the centre $C$ of the circle $x^{2}+y^{2}-2 x-4 y-11=0$ of radius $r$. Let $f_{1}, f_{2}$ be the focal distances of the point C on the ellipse. Then $6 f_{1} {f}_{2}-r$ is equal to

The latus rectum of a conic section is the width of the function through the focus. The positive difference between the length of the latus rectum of 3y = x^2 + 4x - 9 and x^2 + 4y^2 - 6x + 16y - 24 is -

Let $P(2\sqrt{3},\frac{6}{\sqrt{7}})$, $Q$, $R$ 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 $\dfrac{1}{(PQ)^2}+\dfrac{1}{(RS)^2}=\dfrac{p}{q}$, then $p+q$ is equal to

Let $S$ and $S'$ be the foci of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{9}=1$ and $P(\alpha,\beta)$ be a point on the ellipse in the first quadrant. If $(SP)^2+(S'P)^2-SP\cdot S'P=37$, then $\alpha^2+\beta^2$ is equal to:

If $m_1$ and $m_2$ are the slopes of the tangents to the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ which passes through $(5,4)$, then the value of $(m_1 + m_2) - (m_1m_2)$ is equal to

If the maximum distance of normal to the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{b^{2}} = 1$, $b < 2$, from the origin is 1, then the eccentricity of the ellipse is: (1) $\frac{1}{\sqrt{2}}$ (2) $\frac{\sqrt{3}}{2}$ (3) $\frac{1}{2}$ (4) $\frac{\sqrt{3}}{4}$

Let each of the two ellipses $E_1:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, $(a>b)$ and $E_2:\dfrac{x^2}{A^2}+\dfrac{y^2}{B^2}=1$, $(A<B)$ have eccentricity $\dfrac{4}{5}$. Let the lengths of the latus recta of $E_1$ and $E_2$ be $l_1$ and $l_2$, respectively, such that $2l_1^2=9l_2$. If the distance between the foci of $E_1$ is 8, then the distance between the foci of $E_2$ is

A tangent having slope $-\frac{1}{8}$ to the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ intersects the major and minor axes at $A$ and $B$. If $O$ is the origin, then the area of $\triangle OAB$ is

If the chord of the locus of the perpendicular drawn from centre upon any tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is $(x^2 + y^2)^2 = a^2x^2 + b^2y^2$, then $(a - b)$ is equal to

The equations of the directrices of the ellipse 16x2 + 25y2 = 400 are:

For some $\theta \in\left(0, \frac{\pi}{2}\right)$, if the eccentricity of the hyperbola, x2 - y2 sec2 $\theta$ = 10 is $\sqrt{5}$ times the eccentricity of the ellipse, x2 sec2$\theta$ + y2 = 5, then the length of the latus rectum of the ellipse, is:

The minimum area of a triangle formed by any tangent to the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{81} = 1$ and the co-ordinate axes is

If the tangents on the ellipse 4x2 + y2 = 8 at the points. (1, 2) and (a, b) are perpendicular to each other, then a2 is equal to

The equation of the common tangents of the parabola $ y^2 = 4x $ and an ellipse $ \frac{x^2}{4} + \frac{y^2}{3} = 1 $ are -

If y = x and 3y + 2x = 0 are the equations of a pair of conjugate diameters of an ellipse, then the eccentricity of the ellipse 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:

If the foci and vertices of an ellipse be ($\pm$1, 0) and ($\pm$2, 0), then the minor axis of the ellips is:

If foci of an ellipse are (0, ±3) and length of semimajor axis is 5 units, then find the equation of ellipse.

If the normal to the ellipse 3x2 + 4y2 = 12 at a point P on it is parallel to the line, 2x + y = 4 and the tangent to the ellipse at P passes through Q(4, 4) then PQ is equal to

The tangent and normal to the ellipse 3x2 + 5y2 = 32 at the point P(2, 2) meets the X-axis at Q and R, respectively. Then, the area (in sq units) of the $\Delta$PQR is

If the length of the major axis of an ellipse is three times the length of its minor axis, then its eccentricity is

The distance between foci of an ellipse is 6 and the minor axis is 8. Find the eccentricity of the ellipse.

If the distance between the directrices be thrice the distance between the foci, then the eccentricity of an ellipse is:

The position of the point (1, 3) with respect to the ellipse 4x2 + 9y2 - 16x - 54y + 61 = 0 is

The distance between the foci of the ellipse x = 3cos$\theta$, y = 4sin$\theta$ is

Let the equation of ellipse be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $e = \frac{1}{2}$ and equation of directrix is $x = 4$. Then the equation of the ellipse is:

If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is

For each point (x, y) on an ellipse, the sum of the distances from (x, y) to the points (2, 0) and (-2, 0) is 8. Then the positive value of x so that (x, 3) lies on the ellipse is

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point $(-3, 1)$ and has eccentricity $\sqrt{2/5}$ is

Area of the greatest rectangle that can be inscribed in the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ is

If the curves $y^2 = 6x$, $9x^2 + by^2 = 16$ intersect each other at right angles, then the value of $b$ is

The equation of the ellipse whose vertices are ($\pm$5, 0) and foci are ($\pm$4, 0) is

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

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

Question nos. 649 to 651Consider, $E : \dfrac{(x-1)^2}{16} + \dfrac{(y-2)^2}{9} = 1$ and $H : (x-1)^2 - (y-2)^2 = \dfrac{7}{2}$.Column-1 contains equation of tangent to either $E$ or $H$.Column-2 contains image of foci (whose abscissa is greater than 1) of the conic in its tangent.Column-3 contains area (in sq. units) of the triangle formed by joining foci of the conic (according to column-2), its image in the tangent and centre of the conic.Column-1Column-2Column-3(I) $y = x + 6$(i) $(1, \sqrt{7}+2)$(P) $\dfrac{7}{2}$(II) $y = x + 1$(ii) $(-4, \sqrt{7}+7)$(Q) $\dfrac{5\sqrt{7}+7}{2}$(III) $x + y = 3$(iii) $(6, \sqrt{7}-3)$(R) $\dfrac{7}{4}$(IV) $x - y - 4 = 0$(iv) $(1, 2-\sqrt{7})$(S) $\dfrac{5\sqrt{7}-7}{2}$Which of the following options is the only incorrect combination?

Ellipse Questions (292)