The point on the curve

Given that $y^{2}=x$

The tangent makes $45^{\circ}$ angle with $x$-axis.

So, slope of tangent $=\tan 45^{\circ}=1$

$\because$ the point lies on the curve

$\therefore$ Slope of the curve at that point must be 1

$2 y \frac{d y}{d x}=1$

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

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

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

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

So, the correct option is B