Equation of ellipse:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \tag{1}$$
Let $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$ and $D(x_4,y_4)$ be the points of the rectangle such that $A$ is on the director circle, $BD$ is the polar (i.e. the chord) of $A$ and $C$ is the required intersection.
As diagonals of a parallelogram bisect each other,
$$(x_1,y_1)+(x_3,y_3)=(x_2,y_2)+(x_4,y_4)$$
Equation of chord (the polar):
$$\frac{x_1 x}{a^2}+\frac{y_1 y}{b^2}=1$$
$$y=\frac{b^2}{y_1}\left( 1-\frac{x_1 x}{a^2} \right) \tag{2}$$
Substitute $(2)$ into $(1)$,
$$\frac{x^2}{a^2}+\frac{b^2}{y_1^2} \left( 1-\frac{x_1 x}{a^2} \right)^2=1$$
$$\left( \frac{1}{a^2}+\frac{b^2 x_1^2}{a^4 y_1^2} \right)x^2-
\frac{2b^2 x_1}{a^2 y_1^2}x+\frac{b^2}{y_1^2}-1=0$$
Sum of roots:
\begin{align}
x_2+x_4 &= \frac{2a^2b^2 x_1}{a^2 y_1^2+b^2 x_1^2} \\
x_3 &= \frac{2a^2b^2 x_1}{a^2 y_1^2+b^2 x_1^2}-x_1 \\
&= \frac{2a^2b^2-a^2 y_1^2-b^2 x_1^2}{a^2 y_1^2+b^2 x_1^2}x_1 \\
\end{align}
Smilarly,
$$y_3=\frac{2a^2b^2-a^2 y_1^2-b^2 x_1^2}{a^2 y_1^2+b^2 x_1^2}y_1$$
Hence,
$$\frac{y_3}{x_3}=\frac{y_1}{x_1}$$
