SOLUTION: 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 2
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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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