Embibe Experts Solutions for Chapter: Ellipse, Exercise 2: Level 2

$S_1, S_2$ are foci of an ellipse of major axis of length $10$ units and $P$ is any point on the ellipse such that the perimeter of the triangle $P{S}_1{S}_2$ is $15.$ Then eccentricity of the ellipse is

Let $S$ and $S^{\text{'}}$ be two foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. If a circle described on $S{S}^{\text{'}}$ as diameter intersects the ellipse in real and distinct points, then the eccentricity $e$ of the ellipse satisfies

The centre of the ellipse $4x^2+9y^2-16x-54y+61=0$ is

The point of intersection of the tangents at the point $P$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and its corresponding point $Q$ on the auxiliary circle meet on the line

If $\alpha -\beta =$constant, then the locus of the point of intersection of tangents at $P(a\cos \alpha ,b\sin \alpha )$ and $Q\left(a\cos \beta ,b\sin \beta \right)$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

If the tangent at the point $\left(4\cos \theta ,\frac{16}{\sqrt{11}}\sin \theta \right)$ to the ellipse $16x^2+11y^2=256$ is also a tangent to the circle $x^2+y^2-2x=15$, then the value of $\theta$ is

Let $P$ be a point in the first quadrant lying on the ellipse $9x^2+16y^2=144$, such that the tangent at $P$ to the ellipse is inclined at an angle ${135}^{°}$ to the positive direction of $X$-axis. Then, the coordinates of $P$ are

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