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 20111212 16:04:11 