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All the points on the curve $y^{2}=4 a|x+a \sin (x / a)|$, where the tangent is parallel to the axis of $x$ are lies on
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Verified Answer
The correct answer is:
parabola
$y^{2}=4 a\left[x+a \sin \left(\frac{x}{a}\right)\right]$...(i)
$\therefore \quad 2 y \frac{d y}{d x}=4 a\left[1+\cos \left(\frac{x}{a}\right)\right]$...(ii)
If tangent is parallel to $x$ -axis, then $\frac{d y}{d x}=0$
So, from Eq. (i), we get $\cos \left(\frac{x}{a}\right)=-1$
$\therefore$
$$
\sin \left(\frac{x}{a}\right)=0
$$
On putting this value in Eq. (i), we get $y^{2}=4 a(x+0) \Rightarrow y^{2}=4 a x$
So, all the points on the curve
$$
y^{2}=4 a\left(x+a \sin \frac{x}{a}\right)
$$
where the tangent is parallel to the $x$ -axis are lies on parabola.
$\therefore \quad 2 y \frac{d y}{d x}=4 a\left[1+\cos \left(\frac{x}{a}\right)\right]$...(ii)
If tangent is parallel to $x$ -axis, then $\frac{d y}{d x}=0$
So, from Eq. (i), we get $\cos \left(\frac{x}{a}\right)=-1$
$\therefore$
$$
\sin \left(\frac{x}{a}\right)=0
$$
On putting this value in Eq. (i), we get $y^{2}=4 a(x+0) \Rightarrow y^{2}=4 a x$
So, all the points on the curve
$$
y^{2}=4 a\left(x+a \sin \frac{x}{a}\right)
$$
where the tangent is parallel to the $x$ -axis are lies on parabola.
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