Tangent and Normal to an Ellipse

Let $P\left(h,k\right)$ with $h<0, k>0$ be a point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, (where $a>b>0$), whose foci are $F_1$ and $F_2$. The normal at $P$ intersects the major axis at $Q$. If $P{F}_1:P{F}_2=2:1$ then $Q{F}_1:Q{F}_2$ equals

If the quadrilateral formed by four tangents to the ellipse  $\frac{x^2}{9}+\frac{y^2}{4}=1$ is a square, then the area (in sq. units) of the square is equal to

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 $S\left(2\theta \right)$, if the coordinates of $P\left(\theta \right)$ are $(-\sqrt{k},0)$, then find the value of $k$.

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 $S\left(2\theta \right)$, 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 equation of the normal at the point $(16,9)$ to the ellipse $9x^2+16y^2=144$ is 

The equation of the normal at the point $(4,3)$ to the ellipse $9x^2+16y^2=144$ is 

The equation of the normal at the point $(2,-1)$ to the ellipse $9x^2+16y^2=144$ is 

The equation of the normal at the point $(1,-1)$ to the ellipse $9x^2+16y^2=144$ is 

The equation of the normal at the point $(1,3)$ to the ellipse $9x^2+16y^2=144$ is 

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