Question

The point on the curve ${\mathrm{y}}^2=\mathrm{x}$, the tangent at which makes an angle of $45$ degree with $\mathrm{x}-$axis will be given by :

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

Answer 1

The correct option is D

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

Explanation for correct option

Given curve: ${\mathrm{y}}^2=\mathrm{x}$

The slope of the tangent of the curve is $\frac{\mathrm{dy}}{\mathrm{dx}}$

Now on differentiating the given curve, we get;

$\frac{\mathrm{d}\left({\mathrm{y}}^2\right)}{\mathrm{dx}}=\frac{\mathrm{d}\left(\mathrm{x}\right)}{\mathrm{dx}}$

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

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

$\Rightarrow$$\tan \left(45°\right)=\frac{1}{2\mathrm{y}}$ [$\because$Tangent makes $45°$$\Rightarrow$slope of the tangent$=\tan \left(45°\right)$]

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

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

Substituting the above value in the equation of curve, we get;

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

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

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

Hence, the correct option is option(D) i.e. $\left(\frac{1}{4},\frac{1}{2}\right)$