Prove that
the curves *x* = *y* ^{2} and *xy = k* cut at
right angles if 8*k* ^{2} = 1. [**Hint**: Two
curves intersect at right angle if the tangents to the curves at the
point of intersection are perpendicular to each other.]

The equations of the given curves are given as

Putting *x*
= *y*^{2} in *xy* = *k*, we get:

Thus, the point of intersection of the given curves is.

Differentiating
*x* = *y*^{2} with respect to *x*, we have:

Therefore,
the slope of the tangent to the curve *x* = *y*^{2 }atis

On
differentiating *xy* = *k* with respect to *x*, we
have:

∴
Slope of the tangent to the curve *xy = k*^{ }atis
given by,

We know that two curves intersect at right angles if the tangents to the curves at the point of intersection i.e., at are perpendicular to each other.

This implies that we should have the product of the tangents as − 1.

Thus, the given two curves cut at right angles if the product of the slopes of their respective tangents at is −1.

Hence, the
given two curves cut at right angels if 8*k*^{2} = 1.

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