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.]
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 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 8k2 = 1.