An ellipse and hyperbola have the same foci.

Prove that they are orthogonal.

The algebra really got to me so I just did a simple case with a few constants instead of variables.   I decided to let the foci be (±√5,0) and the ellipse and hyperbola be

4x² + 9y² = 36
1x² - 4y² = 4

Solving this system for the first quadrant solution we get x = 6/√5

To prove them orthogonal, next find the slope of each at this point.  For that we need the derivative of each:
y' = -2x/(9√(1 - x²/9)) and y' = x/(4√(x²/4 - 1))
Substitute 6/√5 into each gives
-4/3 and 3/4.

The product of these is 1 and by symmetry, the same will hold true at the other intersections.

Posted by Jer on 2011-12-12 16:04:11