Question

An ellipse and a hyperbola are confocal (having same focus). Length of major axis of ellipse is 8 and length of transverse axis of hyperbola is 4. If P is one of the points of intersection of ellipse and hyperbola then find the product of focal radii of the point P on the ellipse

Answer 1

Answer: 0012

Let $S_1$ and $S_2$ be foci of ellipse as well as hyperbola.
$\text{ Let }P{S}_1=x,P{S}_2=y$
$\because x+y=8\text{......(i) }$
$\text{ and }|x-y|=4\text{.......(ii) }$
$\therefore x=6,y=2\text{ or }x=2,y=6$
$\therefore \text{ Product of focal radii }=xy=12$