Ellipse Questions - JEE

Let $\Delta_1$ be the area of a triangle $PQR$ inscribed in an ellipse and $\Delta_2$ be the area of the triangle $P'Q'R'$ whose vertices are the points lying on the auxiliary circle corresponding to the points $P$, $Q$, $R$ respectively. If the eccentricity of the ellipse is $\dfrac{4\sqrt{3}}{7}$ then the ratio $\dfrac{\Delta_2}{\Delta_1}$ is equal to

The latus rectum of an ellipse is 10 and the minor axis is equal to the distance between the foci. The equation of the ellipse is:

Consider the ellipse E: $\frac{x^{2}}{16}+\frac{v^{2}}{9}=1$, with the major axis AA'. P is a point of E and Q is the corresponding point on the major auxiliary circle. M is the midpoint of PQ. The eccentricity of locus of M:

Let a tangent to the curve $9x^{2} + 16y^{2} = 144$ intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is ____.

$\dfrac{x^2}{r^2 - r - 6} + \dfrac{y^2}{r^2 - 6r + 5} = 1$ will represent the ellipse, if $r$ lies in the interval

Consider an ellipse, whose centre is at the origin and its major axis is along the $x$-axis. If its eccentricity is $3/5$ and the distance between its foci is 6, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is

If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is $\dfrac{3}{2}$ units, then its eccentricity is

An ellipse has eccentricity $e = \dfrac{2}{5}$ (i.e., $1 - e^2 = \dfrac{3}{5}$) and passes through the point \$\left(\dfrac{9}{a^2} + \dfrac{5}{3a^2} = 1\right)$. Find the equation of the ellipse.

Point P divides the length of a staircase in ratio 1 : 2. If the staircase goes from point A on the x-axis to point B on the y-axis, find the locus of point P (i.e., h, k).

If the inclination of the diameter PP' of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ to the major axis is 0 and PP' is the AM of squares of major and minor axes then $\tan\theta$ is equal to

An ellipse having foci $(3, 1)$ and $(1, 1)$ passes through the point $(1, 3)$ has the eccentricity

If a pair of variable straight lines $x^2 + 4y^2 + \alpha xy = 0$ (where $\alpha$ is a real parameter) cut the ellipse $x^2 + 4y^2 = 4$ at two points $A$ and $B$, then the locus of the point of intersection of tangents at $A$ and $B$ is

The equation of ellipse is $3x^2 + 5y^2 = 32$. Find the area of triangle $\triangle ABC$ formed by the tangent and normal to the ellipse at point $(2, 2)$ with the axes (in sq. units).

In an ellipse, the distance between its foci is 6 and minor axis is 8. Then its eccentricity is

An ellipse is sliding along the coordinate axes. If the foci of the ellipse are (1, 1) and (3, 3), then area of the director circle of the ellipse (in sq.units) is

The eccentric angle of a point on the ellipse $x^2 + 3y^2 = 6$ at a distance 2 units from the centre of the ellipse is

If the line $\alpha x+4y=\sqrt{7}$, where $\alpha\in\mathbb{R}$, touches the ellipse $3x^2+4y^2=1$ at the point $P$ in the first quadrant, then one of the focal distances of $P$ is:

An ellipse has its centre at $(1,-2)$, one focus at $(3,-2)$ and one vertex at $(5,-2)$. Then the length of its latus rectum is:

The equation of the chord, of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$, whose midpoint is $(3,1)$ is:

Let $(h,k)$ lie on the circle $C:x^2+y^2=4$ and the point $(2h+1,3k+2)$ lie on an ellipse with eccentricity $e$. Then the value of $\dfrac{5}{e^2}$ is equal to _____.

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

If the angle between the lines joining the end points of minor axis of an ellipse with its foci is $\frac{\pi}{2}$, then the eccentricity of the ellipse is

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

For an ellipse with equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $2e = 1$, find the value of $a^2$ if $b^2 = 32$.

Find the length of major axis of the ellipse with foci S and S′ at (2, 1) and (4, 1) and tangent line x + y = 9.

A line $ y = \dfrac{-2p}{\sqrt{1-p^2}}x + \dfrac{1}{\sqrt{1-p^2}} $ is tangent to an ellipse $ \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 $ for all $ p \in (-1,1) \setminus \{0\} $. Find the eccentricity of the ellipse.

The normal at a variable point $P$ on an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ of eccentricity $e$ meets the axes of the ellipse in $Q$ and $R$. Then the locus of the mid-point of $QR$ is a conic with an eccentricity $e'$ such that:

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) and (4, 1) and a line x + y = 9 is a tangent to this ellipse at point P. Find the eccentricity of the ellipse.

The normal to a curve at $P(x, y)$ meets the $x$-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$, then the curve is a/an

If the tangent at a point P on the parabola $y^{2} = 3x$ is parallel to the line $x + 2y = 1$ and the tangents at the points Q and R on the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{1} = 1$ are perpendicular to the line $x - y = 2$, then the area of the triangle PQR is: (1) $\frac{9}{\sqrt{5}}$ (2) $5\sqrt{3}$ (3) $\frac{3}{2}\sqrt{5}$ (4) $3\sqrt{5}$

Let $A(\alpha,0)$ and $B(0,\beta)$ be the points on the line $5x+7y=50$. Let the point $P$ divide the line segment $AB$ internally in the ratio $7:3$. Let $3x-25=0$ be a directrix of the ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $x$-axis passes through $P$, then the length of the latus rectum of $E$ is equal to

For the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$. Let O be the centre and S and S′ be the foci. For any point P on the ellipse the value of $\frac{PS \cdot PS'}{d^2}$ (where d is the distance of O from the tangent at P) is equal to

Number of perpendicular tangents that can be drawn on the ellipse $\frac{x^2}{16} + \frac{y^2}{25} = 1$ from point (6, 7) is

The normal at the point P on the ellipse $x^2 + 4y^2 = 16$ meets the x-axis at Q. If M is the mid-point of the line segment PQ, then the locus of M intersects the latus rectums of the given ellipse at the points:

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 eccentricity of the ellipse.

Find the eccentricity of an ellipse if the sum of distances from any point P on the ellipse to two foci F₁ at (-4, 4) and F₂ at (3, 3) equals $7\sqrt{2}$, and the distance between foci is $5\sqrt{2}$.

Let the line $y-x=1$ intersect the ellipse $\dfrac{x^2}{2}+y^2=1$ at the points $A$ and $B$. Then the angle made by the line segment $AB$ at the centre of the ellipse is:

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

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 $D_1 - 8D_2 = \frac{8}{5}$

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at (0, 5$\sqrt3$), then the length of its latus rectum is

Find the equation of the director circle of the ellipse $\frac{x^2}{5} + \frac{y^2}{3} = 1$

Let $E_1$ and $E_2$ be two ellipses $\dfrac{x^2}{a^2} + y^2 = 1$ and $x^2 + \dfrac{y^2}{a^2} = 1$ (where $a$ is a parameter). Then the locus of the points of intersection of the ellipses $E_1$ and $E_2$ is a set of curves comprising

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 correct combination?

If the normal at the point $P(\theta)$ to the ellipse $\frac{x^2}{65} + \frac{y^2}{14} = 1$ intersect it again at the point $Q(2\theta)$, then $\cos\theta$ is equal to

Let $P(a\cos\theta_1, b\sin\theta_1)$, $Q(a\cos\theta_2, b\sin\theta_2)$ and $R(a\cos\theta_3, b\sin\theta_3)$ be the vertices of a triangle inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. If $\Delta_1$ = Area of $\triangle PQR$ and $\Delta_2$ = Area of $\triangle P'Q'R'$ where $P', Q', R'$ are the corresponding points on the auxiliary circle, then $\frac{\Delta_1}{\Delta_2} = \frac{1}{7}$. Find the eccentricity of the ellipse.

Which of the following is an interior point of the ellipse $16x^2 + 9y^2 - 16x - 32 = 07$?

The foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{b^2} = 1$ and the hyperbola $\frac{25x^2}{144} - \frac{25y^2}{81} = 1$ coincide. Then the value of $b^2$ is:

Given $4x^2 + y^2 = 8$, let $(a, b)$ be $(\sqrt{2}\cos\theta,\ 2\sqrt{2}\sin\theta)$. The equation of tangent on the given ellipse at point (1, 2) is $4x + 2y = 8$. Find the value (approximately 0.1176) related to the ellipse $\dfrac{x^2}{2} + \dfrac{y^2}{8} = 1$.

Foci of ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$ are given by $(\pm ae, 0)$. The radius of the circle through a focus and having centre at $C(0, 3)$ is:

Ellipse Questions (292)