G Tewani Solutions for Chapter: Ellipse, Exercise 2: DPP

If $\omega$ is one of the angles between the normals to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at two points whose eccentric angles are $\theta$and $\frac{\pi }{2}+\theta$, then $\frac{\left(2\cot \omega \right)}{\sin 2\theta }$ is

From a point on the line $\left(t+2\right)\left(x+y\right)=1, t\ne -2$, tangents are drawn to the ellipse $4x^2+16y^2=1$. It is given that the chord of contact passes through a fixed point. Then the number of integral values of  '$t$'  for which the fixed point always lies inside the ellipse is

At a point $P$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ tangent $PQ$ is drawn. If the point $Q$ be at a distance $\frac{1}{p}$ from the point $P$, where $‘p’$ is distance of the tangent from the origin, then the locus of the point $Q$ is

From a point $P$ perpendicular tangents $\mathrm{PQ}$ and $\mathrm{PR}$ are drawn to ellipse $x^2+4y^2=4$, then locus of circumcentre of the triangle $\mathrm{PQR}$ is

If the normal at any point $P$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the auxiliary circle at $Q$ and $R$, such that $\angle QOR=90°$, where $O$ is the center of the ellipse, then

$P$ and $Q$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $\left(a, b>0\right)$ whose eccentric angles differ by ${90}^{°}$, then

For the ellipses $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$

$AB$ and $CD$ are two equal and parallel chords of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Tangents to the ellipse at $A$ and $B$ intersect at $P$ and tangents at $C$ and $D$ at $Q$. The line $PQ$