+286 An ellipse and a hyperbola have the same foci, $A$ and $B$, and intersect at four points. The ellipse has major axis $24,$ and minor axis $13.$ The distance between the vertices of the hyperbola is $5$. Let $P$ be one of the points of intersection of the ellipse and hyperbola. What is $PA \cdot PB$?
+130462 \( PA \cdot PB \)
Let the center of both be (0,0)
Let the ellipse have its major axis along x....its equation is
x^2 / 12^2 + y^2 / (6.5)^2 = 1
The foci of the ellipse = (+/- sqrt ( 12^2 - 6.5^5] , 0) = ( +/- sqrt (407) / 2) , 0)
For the hyperbola the equation is
x^2/a^2 - y^2/b^2 =1
a^2 = 2.5^2
c = sqrt (407)/2
b^2 = c^2 - a^2 = 407/4 - 2.5^2 = 191/2
The equation of the hyperbola is
x^2/ 2.5^2 - y^2/ (191/2) =1
This is a little difficult to solve but here's a graphical solution using GeoGebra
PA = 9.5
PB =14.5
PA * PB = 9.5 * 14.5 = 137.75
