Question

The point on the curve$y^2=x$, the tangent at which makes an angle $45°$ with $x-$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 for the correct answer:

The tangent makes an angle of $45°$ with the $x-$ axis

Hence, the slope of the tangent is

$m=\tan 45°=1$ $...\left(i\right)$

The slope of the tangent can also be calculated by differentiating the equation of the curve at a point

Equation of the given curve is $y^2=x$

Differentiating with respect to $x$ we get

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

$\Rightarrow$ $\frac{dy}{dx}=\frac{1}{2y}$ $...\left(ii\right)$

From $\left(i\right),\left(ii\right)$ we get

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

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

Substituting this value of $y$ in the equation of the curve we get

$\Rightarrow$${\left(\frac{1}{2}\right)}^2=x$

$\Rightarrow$ $x=\frac{1}{4}$

Hence, the tangent to the curve $y^2=x$ at the point $\left(\frac{1}{4},\frac{1}{2}\right)$ makes an angle of $45°$with the $x-$ axis. So, option $\left(A\right)$ is the correct answer.