Show that the curves 2x = y² and 2xy = k cut at right angles, if k² = 8.

Given:

Curves 2x = y² ...(1)

& 2xy = k ...(2)

We have to prove that two curves cut at right angles if k² = 8

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

⇒ 2x = y²

⇒ 2 = 2y.

⇒

m₁ ...(3)

⇒ 2xy = k

Differentiating above w.r.t x,

⇒ 2(1×) = 0

⇒ = 0

m₂ ...(4)

Since m₁ and m₂ cuts orthogonally,

⇒ ×1

⇒ 1

⇒ x = 1

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

2xy = k & 2x = y²

⇒ (y²)y = k

⇒ y³ = k

⇒ y

Substituting y in 2x = y²,we get,

⇒ 2x = ()²

⇒ 2×1

⇒ 2

⇒ 2³

⇒ 8