Finding the second radius of an ellipse given the first radius and the center

Question

I know the coordinates of A, B and C. A and B are on the axis L1. From that information, I can find the coordinates of the center, the length of radius r1 and the equation of L1 (see picture). Then I can find the equation of L2. What are the steps to find the length of r2? I know it should be possible to find the equation of the curve using the point C and then find the intersection between the curve and L2, but I can't wrap my head around how to do that. An example would be greatly appreciated.

Answer 1

Try the "easy" case first, and put the ellipse in the "standard" position, with the longer axis along the x-axis, and the shorter axis along the y-axis. The equation of the ellipse is now

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

you know $a$, and you know a point $C = (x,y)$.

Answer 2

By what you show, the foci of the ellipse are on line $\;L_1\;$ , say $\;F_1,F_2\;$ , and thus by the definition of ellipse as a locus:

$$|AF_1|+|AF_2|=|BF_1|+|BF_2|=|CF_1|+|CF_2| = k$$

Choose then a point $\;T\;$ on the line $\;L_2\;$ s.t. $\;|TF_1|+|TF_2|=k\;$ and then calculate it's distance to the point of intersection of $\;L_1,L_2\;$ to find out the minor radius

Answer 3

In order to find the equation of $L_2$ translate the intersection of $L_1$ and $L_2$ to the origin, rotate by 90 degrees and then reverse the translation.

In order to find $r_2$ translate to the origin, rotate so the major axis coincided with the x axis and then use @BaronVT 's solution.