the picture depicts a cone and the red line is ellipse's major axis.
I found the major axis to be $\frac{rh^2}{l^3}\sqrt{l^2+3h^2}$ which is calculated using Pythagorean rule, Thales' theorem and Trigonometry.
but I don't know how to find the minor axis.
Is there any way?
Also, I need to find $BC$ segment in order to find the difference between the ellipse's center and the point that the cone's height goes through the ellipse. How come?
Best Answer
Notice that triangles $ARH$, $RHE$, $HEM$, $EMB$, $MBN$ are all similar among them. Hence: $$ HE={h\over l}r,\quad ME={h^2\over l^2}r,\quad BM={h^3\over l^3}r,\quad BN={h^4\over l^4}r. $$ Semi-minor axis is then (see here for a proof): $$ b=\sqrt{EM\cdot BN}={h^3\over l^3}r=BM. $$
Triangles $CMD$ and $CNB$ are similar too, which entails: $$ {BC\over CD}={BN\over DM}={BN\over ME}={h^2\over l^2}. $$ From that, and using your result for $BD$, it is easy to find $BC$.