Show that the following curves intersect orthogonally at the indicated points :

x² = y and x³ + 6y = 7 at (1, 1)

Given:

Curves x² = y ...(1)

& x³ + 6y = 7 ...(2)

The point of intersection of two curves (1,1)

Solving (1) & (2),we get,

First curve is x² = y

Differentiating above w.r.t x,

⇒ 2x

⇒

⇒ m₁ = 2x ...(3)

Second curve is x³ + 6y = 7

Differentiating above w.r.t x,

⇒ 3x² + 6. = 0

⇒

⇒

⇒ m₂ ...(4)

Substituting (1,1) for m₁ & m_2,we get,

m₁ = 2x

⇒ 2×1

m₁ = 2 ...(5)

m₂

⇒

m₂ = ...(6)

when m₁ = 2 & m₂ =

⇒ 2×1

∴ Two curves x² = y & x³ + 6y = 7 intersect orthogonally.