All points on the curve y2=4ax+asin⁡xa  at which the tangents are parallel to the axis of x lie on a

# All points on the curve ${y}^{2}=4a\left(x+a\mathrm{sin}\frac{x}{a}\right)$  at which the tangents are parallel to the axis of $x$ lie on a

1. A

circle

2. B

parabola

3. C

line

4. D

none of these

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### Solution:

We have,

${y}^{2}=4a\left(x+a\mathrm{sin}\frac{x}{a}\right)$                                   ..(i)

For points at which the tangents are parallel to x-axis, we must have

$\begin{array}{l}\frac{dy}{dx}=0\\ ⇒4a\left(1+\mathrm{cos}\frac{x}{a}\right)=0⇒\mathrm{cos}\frac{x}{a}=-1⇒\frac{x}{a}=\left(2n+1\right)\pi \end{array}$

For these values of $x$, we obtain $\mathrm{sin}\frac{x}{a}=0$

Putting $\mathrm{sin}\frac{x}{a}=0$, in (i), we get ${y}^{2}=4ax$

Therefore, all these points lie on the parabola ${y}^{2}=4ax.$

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