17.Show that the curves y2 = 4ax and xy = c2 cut each other orthogonal

1.	17.Show that the curves y2 = 4ax and xy = c2 cut each other orthogonally if c4 = 32 a.
Answer» let x1, y1 be the point of intersection for both the curves y^2 = 4ax => dy/dx = 2a/y = 2a/y1 = m1 (say) from second xy = c^2 => dy/dx = -y/x = -y1/x1 = m2 (say) Now, m1.m2 = -1 (since they intersect orthogonally) => x1 = 2a Again from second, x1y1 = c^2 => y1 = c^2/x [since x1, y1 lies on it] putting in first we get c^4 = 4ax^3 = 4a 8a^3 = 32a^4 or, c4 = 32a4

17.Show that the curves y2 = 4ax and xy = c2 cut each other orthogonally if c4 = 32 a.

Answer»

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

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

from second

xy = c^2 => dy/dx = -y/x = -y1/x1 = m2 (say)

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

=> x1 = 2a

Again from second,

x1y1 = c^2 => y1 = c^2/x [since x1, y1 lies on it]

putting in first we get

c^4 = 4a*x^3 = 4a * 8a^3 = 32a^4

or, c4 = 32a4