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

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