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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

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    $\begingroup$ 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. $\endgroup$ – Jan-Magnus Økland Mar 30 at 7:54
  • $\begingroup$ It is not clear which parameters are fixed and which can vary, in this problem. $\endgroup$ – Aretino Mar 31 at 20:17

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