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I´m interested in solving this problem, preferably by a synthetic method (including conics):

Given an ellipse and two points inside it (other than its foci) construct another ellipse tangent to the given one, using the two points as foci.

I believe there can exist up to four different ellipses to fulfill the condition.

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user131884
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  • What have you tried? Where did you hit your difficulties? What don't you understand? Have you tried simpler problems (e.g., ellipse and circle)? – David G. Stork Apr 24 '20 at 19:37
  • Pick the new foci along the minor axis. Your new ellipse will touch the other one internally at the minor vertices. – Allawonder Apr 24 '20 at 20:03
  • @allawonder The foci are already given. – user Apr 24 '20 at 20:10
  • @user I said take the other two points along the minor axis. – Allawonder Apr 24 '20 at 23:08
  • Perhaps I didnt express propperly. The focii of the new ellipse (b) are the 2 points given inside the given ellipse (a). We must find one point in «a» which should be common to the tangents traced through it to «a» and «b». – user131884 Apr 25 '20 at 12:45
  • Mr. Stock, I really did. In the case of a given circle, there is an equilateral hyperbola centered at the inverse points of the 2 given points which cuts the circle in up to 4 points, solutions to the problem. I've tried to translate the same system to a given ellipse, but no success. – user131884 Apr 25 '20 at 12:55
  • I meant that the hyperbola is centered at the midpoint of the inverses. – user131884 Apr 25 '20 at 13:34

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