Show that the curves 4x = y² and 4xy = k cut at right angles, if k² = 512.

Given:

Curves 4x = y² ...(1)

& 4xy = k ...(2)

We have to prove that two curves cut at right angles if k² = 512

Now ,Differentiating curves (1) & (2) w.r.t x, we get

⇒ 4x = y²

⇒ 4 = 2y.

⇒

m₁ ...(3)

⇒ 4xy = k

Differentiating above w.r.t x,

⇒ 4(1×) = 0

⇒ = 0

m₂ ...(4)

Since m₁ and m₂ cuts orthogonally,

⇒ ×1

⇒ 1

⇒ x = 2

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

4xy = k & 4x = y²

⇒ (y²)y = k

⇒ y³ = k

⇒ y

Substituting y in 4x = y²,we get,

⇒ 4x = ()²

⇒ 4×2

⇒ 8

⇒ 8³

⇒ 512