Given 2 points $F_1$ and $F_2$ and a straight line $l$ which does not cross $[F_1F_2]$. Prove that there exists an unique ellips with $F_1, F_2$ as foci, and tangent $l$. What if $l$ crosses $]F_1,F_2[$?
Would my solution be correct?
And what would would be expected as answer to the extra question?
Answer
Let $\epsilon_1$ (with $a_1, b_1$) be an ellips which satisfies the conditions. Let $\epsilon_2$ (with $a_2,b_2$) be an second ellips which satifies the conditions.
First notice that $c=|OF_1|=|OF_2| = \sqrt{a_1^2-b_1^2} = \sqrt{a_2^2+b_2^2}$ and this means that if $a_1=a_2$ then $b_1=b_2$.
Then there exists 2 situations.
- $\epsilon_1$ and $\epsilon_2$ have a same point in common with the tangent line, say $P$. Which results in $|F_1P|+|F_2P| = 2a_1=2a_2$ and then $a_1 = a_2$.
- If they don't have the same point in commen with the tangent line there is a situation as seen in the picture below.
Then there a certain point $S\in \epsilon_1$ and $S\in \epsilon_2$. So
$$|SF_1|+|SF_2| = 2a_1=2a_2$$
Since $a_1=a_2, b_1=b_2$ these ellipses must be the same.
Extra question
I wouldn't know what to answer, there isn't an ellips which would satisfy the conditions since each $l$ would have 2 points in common with the ellips.
Is there a hyperbola which fits the role then? (I don't think so?)