Prove that the curves x = y² and xy = k cut at right angles* if 8k² = 1.

It is given that the curves x = y² and xy = k

Now, putting the value of x in y = k, we get

y³ = k

Then, the point of intersection of the given curves is

On differentiating x = y² with respect to x, we get

Then, the slope of the tangent at xy = k at

is

As we know that two curves intersect at right angles if the tangents to the curve at the point of intersection are perpendicular to each other.

So, we should have the product of the tangent as -1.

Then, the given two curves cut at right angles if the product of the slopes of their respective tangent at is -1.

⇒ 8k² = 1

Therefore, the given two curves cut at right angles if 8k² = 1.