MHB How to Calculate Distances from Foci to Tangent Line on an Ellipse?

[mathmari]

Hey!

I am looking at the following exercise:

Consider the ellipse $$\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$$
where $p > q > 0$. The eccentricity of the ellipse is $\epsilon =\sqrt{1-\frac{q^2}{p^2}}$ and the points $(\pm \epsilon p, 0)$ on the $x$-axis are called the foci of the ellipse, which we denote by $f_1$ and $f_2$. Verify that $\gamma (t) = (p \cos t, q \sin t)$ is a parametrization of the ellipse. Prove that:

1. The product of the distances from $f_1$ and $f_2$ to the tangent line at any point $p$ of the ellipse does not depend on $p$.
   
2. If $p$ is any point on the ellipse, the line joining $f_1$ and $p$ and that joining $f_2$ and $p$ make equal angles with the tangent line to the ellipse at $p$.

Since $\gamma (t)=(p \cos t , q \sin t)$ we have that $\gamma '(t)=(-p\sin t, q\cos t)$.

The tangent line is given by $l(t)=\gamma (t_0)+t\gamma' (t_0)$.

How can we calculate the distances from $f_1$ and $f_2$ to the tangent line?

Could you give me some hints what we could do at the second question?

 

[I like Serena]

Since $\gamma (t)=(p \cos t , q \sin t)$ we have that $\gamma '(t)=(-p\sin t, q\cos t)$.

The tangent line is given by $l(t)=\gamma (t_0)+t\gamma' (t_0)$.

How can we calculate the distances from $f_1$ and $f_2$ to the tangent line?

Could you give me some hints what we could do at the second question?

Hi mathmari! (Wave)

The distance of a point $\mathbf F$ to a line $l$ through $\mathbf P$ in the direction $\mathbf d$, is given by:
$$d(\mathbf F, l) = ||\boldsymbol\pi_{\mathbf n}(\mathbf F-\mathbf P)|| = (\mathbf F-\mathbf P) \cdot \mathbf n$$
where $\boldsymbol\pi_{\mathbf n}$ is the projection on a normal $\mathbf n$ of the line (perpendicular to $\mathbf d$).
And that projection is given by the dot product if $\mathbf n$ is a vector of unit length.

In your case $\mathbf n$ is given by $(q\cos t, p\sin t)$.

For the second question I think you're supposed to use another application of the dot product.