Ellipse — JEE Main PYQs

If the normal to the ellipse $3x^2+4y^2=12$ 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 $3x^2+5y^2=32$ 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

If tangents are drawn to the ellipse $x^2+2y^2=2$ at all points on the ellipse other than its four vertices than the mid points of the tangents intercepted between the coordinate axes lie on the curve:

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

Let $S$ and $S^{\prime }$ be the foci of an ellipse and $B$ be any one of the extremities of its minor axis. If $\Delta {\mathrm{S}}^{\prime }\mathrm{BS}$ is a right angled triangle with right angle at B and area ($\Delta {\mathrm{S}}^{\prime }\mathrm{BS}$) $=8$ sq. units, then the leng …

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

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

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$, is :

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

An ellipse is drawn by taking a diameter of the circle $(x-1{)}^2+y^2=1$ as its semiminor axis and a diameter of the circle $x^2+(y-2{)}^2=4$ as its semi-major axis. If the centre of the ellipse is the origin and its axes are the coordinate axes, then the equation of the elli …

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

The ellipse $x^2+4y^2=4$ is inscribed in a rectangle aligned with the coordinate axes, which in turn in inscribed in another ellipse that passes through the point $(4,0)$. Then the equation of the ellipse is

A focus of an ellipse is at the origin. The directrix is the line $x=4$ and the eccentricity is $1/2$. Then the length of the semi-major axis is

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

An ellipse has OB as semi minor axis, F and ${\mathrm{F}}^{\prime }$ its focii and the angle ${\mathrm{FBF}}^{\prime }$ is a right angle. Then the eccentricity of the ellipse is

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