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  4. 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

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Q. 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

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Solution:

Let $S_1$ and $S_2$ be foci of ellipse as well as hyperbola.
$\text { Let } PS _1= x , PS _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 }=x y=12 $