Explanation

(b)
The equation of given curve is
$y = \sqrt{x}$ …(i)
$\Rightarrow \frac{dy}{dx} = \frac{1}{2\sqrt{x}}$
Slope of line at $\left( x_{1},\ y_{1} \right),\ m_{1} = \frac{1}{2{\sqrt{x}}_{1}}$
and let line parallel to $x$-axis is $y = k$ …(ii)
Whose slope, $m_{2} = 0$
Since, $45{^\circ}$ is the angle between the line and the curve.
$\therefore\tan{45{^\circ}} = \left| \frac{m_{1} - m_{2}}{1 + m_{1}m_{2}} \right|\ \Rightarrow \ 1 = \left| \frac{\frac{1}{2\sqrt{x_{1}}} - 0}{1} \right|\ \Rightarrow \ x_{1} = \frac{1}{4}$
$\therefore y_{1} = \frac{1}{2}$ [from Eq.(i)]
$\therefore Required\ line\ is\ y = \frac{1}{2}$ [from Eq.(ii)]