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The point on the curve y2 = x, where the tangent makes an angle of \(\rm \frac{\pi}{4}\) with the x-axis, is:
1. (4, 2)
2. \(\left( \frac 1 2, \frac 1 4 \right)\)
3. \(\left( \frac 1 4, \frac 1 2 \right)\)
4. (1, 1)
5. None of these

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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 \(\left( \frac 1 4, \frac 1 2 \right)\).

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