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.