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

Let a tangent be drawn to the ellipse $\frac{x^{2}}{27}+y^{2}=1$at $(3\sqrt{3}\cos\,\theta\,,\sin\,\theta\,)$ where $\theta\,\in\,\left(0,\frac{\pi\,}{2}\right)$. Then thevalue of $\theta\,$ such that the sum of…

Two sets $A$ and $B$ are as under:$A=\left{\left(a,b\right)\in\,R\times\,R:\left|a-5\right|<1\text{and}\left|b-5\right|<1\right};$…

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…

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:

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…

On the ellipse $\frac{x^{2}}{8}+\frac{y^{2}}{4}=1$,let $P$ be a point in the second quadrant such that the tangent at $P$ to the ellipse is perpendicular to the line $x+2y=0$. Let $S$ and $S^{'}$ be the foci of the…

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$S$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 $O\left(0,0\right)$ and $A\left(0,1\right)$be two fixed points. Then, the locus of a point $P$ such that the perimeter of $\Delta\,AOP$ is $4$ is

Let $A=\{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R}:|\alpha-1| \leq 4 \text { and }|\beta-5| \leq 6\}$ and $B=\left\{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R}: 16(\alpha-2)^2+9(\beta-6)^2 \leq 144\right\}$

Let the maximum area of the triangle that can be inscribed in the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{4}=1,a>2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel…