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 $PA \cdot PB$?
500
Stop Cheating! Thankfully, the answer isn't 600.
To find $PA \cdot PB$, we will first need to find the coordinates of the points $A$, $B$, and $P$.
Let's start by determining the center and major and minor axes of the ellipse. The major axis of the ellipse has a length of 50, which means it extends 25 units in both directions from the center. The minor axis has a length of 40, which means it extends 20 units in both directions from the center.
Since the ellipse and the hyperbola have the same foci, we can conclude that the distance between the center of the ellipse and the foci $A$ and $B$ is equal. Let the distance between the center of the ellipse and $A$ (or $B$) be denoted as $a$.
Now, let's find the coordinates of $A$, $B$, and $P$.
Since $PA$ and $PB$ are radii of the ellipse, we know that $PA = \frac{50}{2} = 25$ and $PB = \frac{40}{2} = 20$. So, $PA \cdot PB = 25 \cdot 20 = 500$.