Question 1077688: An ellipse and a hyperbola have the same foci, A and B, and intersect at four points. The ellipse has major axis 50, and minor axis 40. The hyperbola has conjugate axis of length 20. Let $P$ be a point on both the hyperbola and ellipse. What is PA*PB?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! The ellipse has
 (the semi-major axis), and
 (the semi-minor axis).
The focal distance,  , can be found from
 .
Substituting known values,  --> -->  .
So, it looks like this  (maybe shifted and/or rotated).
According to the definition of ellipse,
if P is a point of an ellipse with foci A and B,
 .
The hyperbola has
, just like the ellipse,
and has  .
Knowing that in a hyperbola
 , we can find  .
In a hyperbola with foci A and B, for any point P, by definition
 .
Squaring both sides in the equations found involving the distances PA and PB,
 --> and
 ---> .
Subtracting from , we get
 -->  --> 
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