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The foci of an ellipse are S,S’ and P,P’ are two points on the curve on opposite sides of the major axis.SP’ meets S’P at Q’, and S’P’ meets SP at Q. To prove that Q and Q’ lie on another ellipse with foci S,S’. I am uncertain whether the best approach is geometric,or analytic. Any advice would be appreciated.

D. Spencer
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  • What are your thoughts? Any sketch made? A hand sketch is necessary for an analytical geometry solution. Which point marks the common focus? – Narasimham Nov 09 '20 at 15:52
  • I have shown,by both geometric and analytic methods that the lines TP and TP',where T is the intersection of the tangents at P and P',subtend equal angles at the focus S (this is a well-known result).It can easily be shown that the lines QT and QT' also subtend equal angles.It does not seem clear that we can reverse the arguments, either geometrically or analytically to prove the desired result.I have taken S as the focus in question,and have tried to incorporate S' by means of equal reflection at P etc,but to no avail.The common foci in the solution are both S and S'. – D. Spencer Nov 09 '20 at 23:12

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HINT1:

How to manipulate between ellipses of constant inter-focal distance $SS'= 2c$?

$$\dfrac{x_1^2}{a_1^2}+\dfrac{y_1^2}{a_1^2-c^2}\tag1$$ $$\dfrac{x_2^2}{a_2^2}+\dfrac{y_2^2}{a_2^2-c^2}\tag2$$

HINT2:

Imagine a laser shot in the direction $SQP.$ How does it get reflected/ricocheted at $Q$ and $P$?

Narasimham
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  • I understand the respective equations of the confocal ellipses and also the reflective property.I know there areconfocal ellipse passing through Q and Q’ respectively,but cannot prove they are identical. – D. Spencer Nov 10 '20 at 23:28
  • Let the tangents at P,P' meet at T. Join QT and Q'T. We can construct a circle centre T touching SP,SP,S'P and S'P'. By considering the reflective properties of the ellipse we can show that QT,Q'T are tangents at Q,Q' respectively. We have QS'T=Q'S'T so the ellipses at Q and Q' are identical and the result is proved. Thank you for your advice on this. – D. Spencer Nov 13 '20 at 16:43