if the curves y²=4ax and xy=c² cut at right angles ,prove that

C⁴=32a⁴

let x1,y1 be the point of intersection for both the curves

y² = 4ax => dy/dx = 2a/y = 2a/y1 = m1 (say)

from second

xy = c² => dy/dx = -y/x = -y1/x1 = m2 (say)

Now, m1.m2 = -1 (since they intersect orthogonally)

=> x1 = 2a

Again from second,

x1y1 = c² => y1 = c²/x  [since x1, y1 lies on it]

putting in first we get

c⁴ = 4ax₁³ = 4a x 8a³ = 32 a⁴

or, c⁴ = 32a⁴ done... !!