Question

The point on the curve $y^2 = x$ where the tangent makes an angle of $\pi /4$ with X-axis is

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

Answer 1

Answer: (B)

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

Answer 2

We have, $y^{2}=x\,\,\,\,\,\dots(i)$
$\therefore 2 y \frac{d y}{d x}=1$
$\Rightarrow \frac{dy}{dx}=\frac{1}{2 y}$
$\therefore$ Slope of tangent $=\frac{1}{2 y}\,\,\,\,\,\,\dots(ii)$
Now, tangent makes an angle of $\pi / 4$ with $X$ -axis
$\therefore$ Slope of tangent $=\tan \frac{\pi}{4}=1\,\,\,\,\,\dots(iii)$
From Eqs. (ii) and (iii), we get
$\frac{1}{2 y}=1$
$\Rightarrow y=\frac{1}{2}$
Putting, $y=\frac{1}{2}$ in E (i), we get
$\left(\frac{1}{2}\right)^{2} =x$
$\Rightarrow \, x=\frac{1}{4}$
$\therefore$ Required point is $\left(\frac{1}{4}, \frac{1}{2}\right)$