Question

The point on the curve $y^2=x$, the tangent at which makes an angle $45°$ with the $x$ axis is

• A. $\left(\frac{1}{4},\frac{1}{2}\right)$
• B. $\left(\frac{1}{2},\frac{1}{4}\right)$
• C. $\left(\frac{1}{2},-\frac{1}{2}\right)$
• D. $\left(\frac{1}{2},\frac{1}{2}\right)$

Answer 1

The correct option is A

$\left(\frac{1}{4},\frac{1}{2}\right)$

Explanation of the correct option.

Compute the required point:

Given: The tangent makes an angle $45°$ with the $x$ axis

Since slope is given by $\tan \left(\theta \right)$

Therefore, the slope of the tangent $m=\tan \left(45°\right)$

Given: $y^2=x$……………..$\left(1\right)$

Differentiate with respect to $x$,

$2y\frac{dy}{dx}=1$

$\Rightarrow$ $\frac{dy}{dx}=\frac{1}{2y}$

Since $\frac{dy}{dx}$ is the slope.

$\Rightarrow \tan \left(45°\right)=\frac{1}{2y}$

$\Rightarrow$ $1=\frac{1}{2y}$

$\Rightarrow$ $y=\frac{1}{2}$

Substitute the value of $y$ in equation $\left(1\right)$,

$$\begin{gathered} {\left(\frac{1}{2}\right)}^2=x \\ \Rightarrow x=\frac{1}{4} \end{gathered}$$

Hence, the point is $\left(\frac{1}{4},\frac{1}{2}\right)$.

Hence, option $\left(A\right)$ is the correct answer.