'le conical sections' (1 Viewer)

[dawso]

p(acos@, bsin@) and Q(acos(-@), bsin(-@)) are the extremities of the latus rectum x=ae of the ellipse (insert standard ellipse equation here...)

show that PQ has length 2b^2/a

i can kinda work it out, but basically cause i know the result, anyone got a good proof?

 

[KFunk]

x²/a² + y²/b² = 1 , x=ae
e² + y²/b² = 1
y = ±√(b²(1 - e²))
=±b√(1 - e²)
P and Q lie on a vertical line hence the distance between them is |y_p - y_q|

|PQ| = 2b√(1 - e²) remember that b² = a²(1 - e²) hence (1 - e²) = b²/a²

|PQ| = 2b√(b²/a²) = 2b²/a