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Question :

An ellipse whose length of semi-major axis is $l_1$, and a confocal hyperbola with length of semi-transverse axis $l_2$ meet at P. If S and S' are the foci then prove that $(SP)(S'P) = l_1^2-l_2^2$

I assumed the semi-minor axis of ellipse as $b_1$ and semi-conjugate axis of hyperbola as $b_2$, formed the equation of the hyperbola and ellipse, solved them for the intersection point, calculated the eccentricities, found out the coordinates of the foci, and evaluated $(SP)(S'P)$. I ended up with a mess of variables, nowhere close to the answer. I am sure my method will not lead to the answer.. Looking for an easier method. Thanks for any help.

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  • $\begingroup$ Hint: Recall the definition of both the conics. It's something related to sum and difference of distances of the points from their foci. $\endgroup$ – SirXYZ May 2 '17 at 17:00
  • $\begingroup$ @SirXYZ Yes I know that for the ellipse we have $SP+S'P=2l_1$ and for the hyperbola it is $|SP-S'P|=2l_2$ $\endgroup$ – Shreyas S May 5 '17 at 3:24
  • $\begingroup$ @SirXYZ Oh okay I got it we add and subtract the above equations and multiply them to get the result. Thank you $\endgroup$ – Shreyas S May 5 '17 at 3:25
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Hint: Well, $$SP+S'P=\frac{2l_1}{\sqrt{e_{ellipse}^2-1}}$$ via Property of ellipse

and $$SP-S'P=\frac{2l_2}{\sqrt{e_{hyperbola}^2-1}}$$ via property of hyperbola

and $$\frac{e_{ellipse}\times l_1}{\sqrt{e_{ellipse}^2-1}}=\frac{e_{hyperbola}\times l_1}{\sqrt{e_{hyperbola}^2-1}}$$ via confocal nature.

Hope it helps! Cheers

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