If $P$ be any point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ whose ordinate is $y\text{'}$, the angle between the tangent at $P$ and the focal distance of $P$ is

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

The equation of tangent at $P$ is $\frac{x{x}^{\mathrm{\prime }}}{a^2}+\frac{y{y}^{\mathrm{\prime }}}{b^2}=1$

The slope of the tangent is $-\frac{b^2{x}^{\mathrm{\prime }}}{a^2{y}^{\mathrm{\prime }}}$ and the slope of $\mathrm{SP}$ is $\frac{y^{\mathrm{\prime }}}{x^{\mathrm{\prime }}+ae}$

Now,

 $\begin{array}{l}\tan \theta =\frac{\frac{y^{\mathrm{\prime }}}{x^{\mathrm{\prime }}+ae}+\frac{b^2{x}^{\mathrm{\prime }}}{a^2{y}^{\mathrm{\prime }}}}{1-\left(\frac{y^{\mathrm{\prime }}}{x^{\mathrm{\prime }}+ae}\cdot \frac{b^2{x}^{\mathrm{\prime }}}{a^2{y}^{\mathrm{\prime }}}\right)}\\ =\frac{a^2{b}^2+b^2{x}^{\mathrm{\prime }}ae}{x^{\mathrm{\prime }}{y}^{\mathrm{\prime }}{a}^2{e}^2+a^{2}e{y}^{\mathrm{\prime }}}\\ =\frac{b^{2}a\left(a+e{x}^{\mathrm{\prime }}\right)}{a^{2}e{y}^{\mathrm{\prime }}\left(a+e{x}^{\mathrm{\prime }}\right)}\\ =\frac{b^2}{ae{y}^{\mathrm{\prime }}}\end{array}$

$\Rightarrow \theta ={\tan }^{-1}\left(\frac{b^2}{ae{y}^{\mathrm{\prime }}}\right)$.