Correct Answer - Option 3 :
\(\left( \frac 1 4, \frac 1 2 \right)\)
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 \(\left( \frac 1 4, \frac 1 2 \right)\).