# An ellipse and a confocal hyperbola meet at P ..

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.

• Hint: Recall the definition of both the conics. It's something related to sum and difference of distances of the points from their foci. Commented May 2, 2017 at 17:00
• @SirXYZ Yes I know that for the ellipse we have $SP+S'P=2l_1$ and for the hyperbola it is $|SP-S'P|=2l_2$ Commented May 5, 2017 at 3:24
• @SirXYZ Oh okay I got it we add and subtract the above equations and multiply them to get the result. Thank you Commented May 5, 2017 at 3:25

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.