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: 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