Ellipse

Let $C$ be the largest circle centred at $\left(2,0\right)$ and inscribed in the ellipse$\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$. If$\left(1,\alpha\,\right)$lies on $C$, then $10\alpha\,^{2}$is equal to ______

A ray of light through $\left(2,1\right)$is reflected at a point $P$ on the $y-$ axis and then passes through the point$\left(5,3\right).$If this reflected ray is the directrix of an ellipse with eccentricity…

Let $E_{1}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,a>b.$Let $E_{2}$ be another ellipse such that it touches the end points of major axis of $E_{1}$ and the foci of $E_{2}$ are the end points of minor axis of…

Let$P$ be a parabola with vertex$\left(2,3\right)$and directrix$2x+y=6$.Let an ellipse$E:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,a>b$ofeccentricity$\frac{1}{\sqrt{2}}$pass through the focus of the parabola $P$. Then…

Let$S$and$S^{'}$ be the foci of an ellipse and$B$be any one of the extremities of its minor axis. If $\Delta\,S^{'}BS$ is a right angled triangle with right angle at$B$and area…

Let an ellipse with centre$\left(1,0\right)$and latus rectum of length $\frac{1}{2}$ have its major axis along x-axis. If its minor axis subtends an angle $60^{\circ\,}$ at the foci, then the square of the sum of the…

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 $\left(0,5\sqrt{3}\right),$ then the length of its latus rectum is:

Consider an ellipse, whose center is at the origin and its major axis is along the $x$-axis. If its eccentricity is $\frac{3}{5}$ and the distance between its foci is$6$, then the area (in sq. units) of the…

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…

Let$PQ$ be a focal chord of the parabola $y^{2}=4x$such that it subtends an angle of $\frac{\pi\,}{2}$at the point $\left(3,0\right)$. Let the line segment$PQ$ be also a focal chord of the ellipse…

The line $y=x+1$meets the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$at two points $P$ and $Q$. If$r$ is the radius of the circle with $PQ$as diameter then $\left(3r\right)^{2}$is equal to

If the points of intersection of two distinct conics $x^{2}+y^{2}=4b$and $\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1$lie on the curve $y^{2}=3x^{2}$, then $3\sqrt{3}$ times the area of the rectangle formed by the…