The point on the curve y² = x, where the tangent makes an angle of \(\rm \frac{\pi}{4}\) with the x-axis, is:

Concept:

The angle made by the tangent to the curve y = f(x) at a point (a, b), with the x-axis, is given by m = tan θ = \(\rm \left[\frac{dy}{dx}\right]_{(a, b)}\).

Calculation:

The given curve is y² = x.

⇒ \(\rm 2y\frac{dy}{dx}=1\)

⇒ \(\rm \frac{dy}{dx}=\frac{1}{2y}\)

For the tangent to make an angle of \(\rm \frac{\pi}{4}\), we must have:

tan \(\rm \frac{\pi}{4}\) = \(\rm \frac{dy}{dx}=\frac{1}{2y}\)

⇒ 1 = \(\rm \frac{1}{2y}\)

⇒ y = \(\rm \frac{1}{2}\)

Also, x = y² = \(\rm \frac{1}{4}\).

The required point is \(\left( \frac 1 4, \frac 1 2 \right)\).