# Finding the condition so that length of a common tangent to an ellipse and another concentric ellipse is maximum

I am trying to find the condition so that for an ellipse $$E_1$$ and another ellipse $$E_2$$, if $$A$$ is a point on $$E_1$$ and $$B$$ is a point on $$E_2$$, and tangent to both $$E_1$$ and $$E_2$$ passes through $$AB$$. What would be the condition such that $$AB$$ is maximum?

For starters, we can write the equation of tangent on each ellipse, $$E_1$$ has semi-major and semi-minor axes as $$a_1$$ and $$b_1$$, and $$E_2$$ has semi-major and semi-minor axes as $$a_2$$ and $$b_2$$ respectively. Equation of tangents on $$E_1$$, say $$T_1=0$$ would be $$y=mx\pm \sqrt{a_1^2m^2+b_1^2}$$ and on $$E_2$$, say $$T_2=0$$ would be $$y=mx\pm\sqrt{a_2^2m^2+b_2^2}$$. Also since these equations represent the same line, then $$a_1^2m^2+b_1^2=a_2^2m^2+b_2^2$$.

How to proceed? Any hints are appreciated. Thanks

• See this. Taking $a_1=b_2, a_2=b_1, a_1=\frac1{a_2}$ and let $a_1$ grow without bound I think you can get arbitrarily large common tangent segments. – Jan-Magnus Økland Mar 30 at 7:54
• It is not clear which parameters are fixed and which can vary, in this problem. – Aretino Mar 31 at 20:17