Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1 - Math - Application of Derivatives

Question

Prove that the curves x = y 2 and xy
= k cut at right angles if 8k 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 ]

Gopal Mohanty · Accepted Answer

The equations of the given curves are given as

Putting x = y2 in xy = k, we get:

Thus, the point of intersection of the given curves is

Differentiating x = y2 with respect to x, we have:

Therefore, the slope of the tangent to the curve x = y2 at

is

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

∴ Slope of the tangent to the curve xy = k at

is 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 8k2 = 1

Prove that the curves x = y 2 and xy = k cut at right angles if 8k 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.]

Prove that the curves x = y ² and xy = k cut at right angles if 8k 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.]