Question

All the points on the curve $y=\sqrt{x+\sin x}$ at which the tangent is parallel to $x$ - axis lie on a/an

• A. straight line
• B. circle
• C. parabola
• D. ellipse

Answer 1

Answer: (C)

$y=\sqrt{x+\sin x}$
Let at $P\left(x_1,y_1\right)$ tangent be parallel to $x$ -axis.
So ${\left(\frac{dy}{dx}\right)}_{\left(x_1,y_1\right)}=0$
$\Rightarrow \frac{1+\cos x_1}{2\sqrt{x_1+\sin x_1}}=0$
$\Rightarrow 1+\cos x_1=0$
$\Rightarrow \cos x_1=-1\Rightarrow x_1=\pi \Rightarrow \sin x_1=0\dots (i)$
Since $\left(x_1,y_1\right)$ lies on given curve,
$y_1^2=x_1+\sin x_1$
$\Rightarrow y_1^2=x_1$
$(\sin x_1=0$ from (i))
$\therefore$ Locus of $\left(x_1,y_1\right)$ is $y^2=x$
which is a parabola.