#### Question

The point on the curve y^{2} = x where tangent makes 45° angle with x-axis is

(a) (1/2, 1/4)

(b) (1/4, 1/2)

(c) (4, 2)

(d) (1, 1)

#### Solution

(b) (1/4, 1/2)

Let the required point be (*x*_{1}, y_{1}).

The tangent makes an angle of 45^{o} with the *x*-axis.

∴ Slope of the tangent = tan 45^{o} = 1

\[\text { Since, the point lies on the curve } . \]

\[\text { Hence, } {y_1}^2 = x_1 \]

\[\text { Now,} y^2 = x\]

\[ \Rightarrow 2y\frac{dy}{dx} = 1\]

\[ \Rightarrow \frac{dy}{dx} = \frac{1}{2y}\]

\[\text { Slope of the tangent } = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =\frac{1}{2 y_1}\]

\[\text { Given }:\]

\[\frac{1}{2 y_1} = 1\]

\[ \Rightarrow 2 y_1 = 1\]

\[ \Rightarrow y_1 = \frac{1}{2}\]

\[\text{ Now,} \]

\[ x_1 = {y_1}^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4}\]

\[ \therefore \left( x_1 , y_1 \right) = \left( \frac{1}{4}, \frac{1}{2} \right)\]