Question

The line which is parallel to x-axis and crossed the curve $y=\sqrt{x}$ at an angle ${45}^{\circ }$, is

• A. $y=\frac{1}{4}$
• B. $y=\frac{1}{2}$
• C. $y=1$
• D. $y=4$

Answer 1

The correct option is B $y=\frac{1}{2}$

The equation of given curve is
$y=\sqrt{x}$....(i)
$\Rightarrow \frac{dy}{dx}=\frac{1}{2\sqrt{x}}$
Slope of line at $(x_1,y_1),m_1=\frac{1}{2\sqrt{x_1}}$
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.
$\therefore \tan {45}^{\circ }=\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$
$\Rightarrow 1=\left|\frac{\frac{1}{2\sqrt{x_1}}-0}{1}\right|\Rightarrow x_1=\frac{1}{4}$
$\therefore$ From Eq. (i), $y_1=\frac{1}{2}$
$\therefore$ Equation of line is
$y=\frac{1}{2}$ [ from Eq. (i) ]