Question

If the curves ay + x² = 7 and y = x³ cut each other orthogonally at a point, find a.

Answer 1

Given curves are ay + x² = 7 and y = x³ ...(i)

⇒ a(dy/dx)  + 2x = 0, dy/dx = 3x²

⇒ dy/dx = – 2x/a , dy/dx = 3x²

Since the curves cut orthogonally, so

⇒ – 2x/a × 3x² = –1

⇒ a = 6x³ ...(ii)

On solving (i) and (ii), we get,

6x⁶ + x² = 7

⇒ x = 1

Hence, the value of a is 6 .