The point on the curve y^2=x where tangent makes 45° angle with x-axi…

The correct answer is Option (2) - $\left(\frac{1}{4}, \frac{1}{2}\right)$

$y^2=x$  ...(I)

so differentiating wrt x

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

when tangent makes 45° with x axis

$⇒\frac{dy}{dx}=\tan 45°=1$

$⇒\frac{1}{2y}=1$ so $y=\frac{1}{2}$

from (1) → $x=\frac{1}{4}$

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