Tangent and Normal to an Ellipse

If a line $px+qy=12$ is a tangent to the ellipse $4x^2+9y^2=16$, then $pq$ can not be equal to :

Find coordinate point of contact of tangent $-4x+10y=25$ to an ellipse $\frac{x^2}{25}+\frac{y^2}{4}=1$

Find coordinate point of contact of tangent $-3x+10y=25$ to an ellipse $\frac{x^2}{25}+\frac{y^2}{4}=1$

The angle is subtended by common tangents of two ellipses $4(x-6{)}^2+25y^2=100$ and $4(x+3{)}^2+y^2=4$ at the origin.

The angle is subtended by common tangents of two ellipses $4(x-5{)}^2+25y^2=100$ and $4(x+2{)}^2+y^2=4$ at the origin.

Find the angle is subtended by common tangents of two ellipses $4(x-4{)}^2+25y^2=100$ and $4(x+1{)}^2+y^2=4$ at the origin.

If the normal at the point $P\left(\theta \right)$ to the ellipse $\frac{x^2}{12}+\frac{y^2}{8}=1$ intersects it again at the point $Q\left(2\theta \right)$, then find the coordinates of $P\left(\theta \right)$.

If the normal at the point $P\left(\theta \right)$ to the ellipse $\frac{x^2}{6}+\frac{y^2}{4}=1$ intersects it again at the point $Q\left(2\theta \right)$, then find the coordinates of $P\left(\theta \right)$.

If the normal at the point $P\left(\theta \right)$ to the ellipse $\frac{x^2}{3}+\frac{y^2}{2}=1$ intersects it again at the point $Q\left(2\theta \right)$, then find the coordinates of $P\left(\theta \right)$.

If the normal at the point $P\left(\theta \right)$ to the ellipse $\frac{x^2}{14}+\frac{y^2}{5}=1$ intersects it again at the point $Q\left(2\theta \right)$, then find the coordinates of $P\left(\theta \right)$.

The number of maximum normals that can be drawn from any point to an ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ is $3$.

The equation of the normal at the point $(2,3)$ to the ellipse $9x^2+16y^2=144$ is 

The equation of the normal at the point $(2,3)$ to the ellipse $9x^2+16y^2=180$ is 

The number of maximum normals that can be drawn from any point to an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

The number of distinct normals that can be drawn from the point $(0,6)$ to the ellipse $\frac{x^2}{169}+\frac{y^2}{25}=1$.

Find the equation of the chord joining two points $P(6\cos \theta ,5\sin \theta )\text{ and }Q(6\cos \varphi ,5\sin \varphi )$ on the ellipse.

Find the equation of the chord joining two points $P(5\cos \theta ,3\sin \theta )\text{ and }Q(5\cos \varphi ,3\sin \varphi )$ on the ellipse.

Find the equation of the chord joining two points $P(4\cos \theta ,3\sin \theta )\text{ and }Q(4\cos \varphi ,3\sin \varphi )$ on the ellipse.

Find the equation of the chord joining two points $P(3\cos \theta ,2\sin \theta )\text{ and }Q(3\cos \varphi ,2\sin \varphi )$ on the ellipse.

Prove that, the equation of the chord joining two points having eccentric angles $\theta \text{ and }\varphi$ on the ellipse is $\frac{x}{a}\cos \left(\frac{\theta +\varphi }{2}\right)+\frac{y}{b}\sin \left(\frac{\theta +\varphi }{2}\right)=\cos \left(\frac{\theta -\varphi }{2}\right)$.