Theory of tangents and normals of an ellipse

Question

What are the number of distinct normals that can be drawn to an ellipse from a point inside ,on and outside an ellipse?

Answer 1

WLOG we can assume the equation of the ellipse to be $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\ \ \ \ (1)$$ whose any point can be written as $(a\cos t,b\sin t)$

So, the gradient of the line passing through $(h,k)$ and $(a\cos t,b\sin t)$ is $\dfrac{k-b\sin t}{h-a\cos t}\ \ \ \ (2)$

The gradient of the tangent of $(1)$ at $(a\cos t,b\sin t)$ is $-\dfrac{b\cos t}{a\sin t}$

The gradient of the normal of $(1)$ at $(a\cos t,b\sin t)$ is $\dfrac{a\sin t}{b\cos t}\ \ \ \ (3)$

Equating $(2),(3);$

$\dfrac{k-b\sin t}{h-a\cos t}=\dfrac{a\sin t}{b\cos t}$

$\implies\dfrac{k-b\sin t}{a\sin t}=\dfrac{h-a\cos t}{b\cos t}=\lambda$(say)

$\implies\sin t=\dfrac k{a\lambda+b},\cos t=\dfrac h{a+b\lambda}$

$\implies\left(\dfrac k{a\lambda+b}\right)^2+\left(\dfrac h{a+b\lambda}\right)^2=1$

On rearrangement, this gives a biquadratic Equation in $\lambda$

So, four normals can be drawn from a given point $(h,k)$ to $(1)$