Question

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

x² = y and x³ + 6y = 7 at (1, 1)

Answer 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 \(=\frac{dy}{dx}\)

\(\Rightarrow\frac{dy}{dx}\) = 2x

⇒ m₁ = 2x ...(3)

Second curve is x³ + 6y = 7

Differentiating above w.r.t x,

Substituting (1,1) for m₁ & m₂,we get,

m₁ = 2x

⇒ 2×1

m₁ = 2 ...(5)

⇒ \(2\times\frac{-1}{2}=-1\)

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