ratio of tangent to the ellipse

Question

The tangent at point $P = ( a \cos \phi, b \sin \phi)$ on the ellipse

$\frac{x^2} {a^2} + \frac{y^2}{b^2}=1$

meets the $x$ and $y$ axes at the points $X$ and $Y$, respectively.

Find in terms of $\phi$ the ratio $\frac{|PX|}{|YP|}$

Answer 1

The tangent line at $P$ has equation $$ \frac{x}{a}\cos\phi + \frac{y}{b}\sin\phi = 1 $$ Substituting $y=0$, we find that the point $X$ has coordinates $\left(\tfrac{a}{\cos\phi}, 0 \right)$.

Similarly, $Y$ has coordinates $\left(0, \tfrac{b}{\sin\phi}\right)$.

Now just figure out the distances $PX$ and $PY$, and divide to get the desired ratio.