Lines intersections distance on the asymptotes

Question

Like in picture we have two lines. Lenght of one of them is 2E and other's lenght 2C and also ellipse asymptotes are A and B and its center is on origin(0,0) I want to find D and F How can I calculate these lenghts?

Answer 1

Assuming $A$ and $B$ are the semi-axes AND the point with coordinates $(x,y) = (C, E)$ lies on the ellipse, which is given by $$ \left( \frac{x}{A} \right)^2 + \left( \frac{y}{B} \right)^2 = 1 $$ so $$ \left( \frac{C}{A} \right)^2 + \left( \frac{E}{B} \right)^2 = 1 $$ holds, it is simply $$ F = A - C = A - A \sqrt{1 - (E/B)^2} \\ D = B - E = B - B \sqrt{1 - (C/A)^2} $$