Question

The points on the curve $y^2=4a(x+a\sin \frac{x}{a})$ at which the tangent is parallel to x axis lie on -

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

Answer 1

The correct option is B a parabola

$y^2=4a(x+a\sin \frac{x}{a})$
$\Rightarrow 2y\frac{dy}{dx}=4a(1+a\cos \frac{x}{a}\times \frac{1}{a})$
$\Rightarrow \frac{dy}{dx}=\frac{2a}{y}(1+\cos \frac{x}{a})$
Now for tangent to be parallel to x-axis,
$\frac{dy}{dx}=0=\frac{2a}{y}(1+\cos \frac{x}{a})$
$\Rightarrow \frac{x}{a}=\pi$
$\Rightarrow x=a\pi$
$\therefore y^2=4a(a\pi +a\times 0)$
$\Rightarrow y^2=4a^2\pi =4ax$
Clearly, this point lie on a parabola.
Hence, option 'B' is correct.