On an ellipse $E$ with vertex $P$ and $P'$ on the major axis and vertex $Q$ and $Q'$ on the minor axis. Choose $R(x_1,y_1)$, the reflection of $R$ on the major axis is $R'$ and on the minor axis is $R''$. Define the perpendicular projection of the intersection point of $PR$ and $P'R'$ on the major axis. And define the perpendicular projection of the intersection point of $QR$ and $Q'R''$ on the minor axis. Prove that the line drawn from these two projections is the tangent of $R$.
I get stuck every time, sorry if the answer is obvious, I'm only 16. I attached a quick sketch I made. This is the sketch
Let the point $R$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be $(a\cos\theta,b\sin\theta)$. It's obvious that $R'=(a\cos\theta,-b\sin\theta)$ and $R''=(-a\cos\theta,b\sin\theta)$.
Now, just find the intersection point of $PR, P'R'$ and $QR, Q'R''$. Then find their respective projections $C$ and $D$.
Equation of line $CD$ comes out to be $$\frac{x\cos\theta}{a}+\frac{y\sin\theta}{b}=1$$ Note that it passes through $R$. It is tangent to the ellipse since solving the line with ellipse gives only one solution i.e. $(x,y)=(a\cos\theta,b\sin\theta)$.