All the points on the curve y^{2}=4a|x+a sin( | vedclass

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(B) Given the curve equation: $y^{2}=4a|x+a \sin(x/a)|$. For the tangent to be parallel to the $x$-axis,the derivative $\frac{dy}{dx}$ must be $0$. Co

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parabola

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(B) Given the curve equation: $y^{2}=4a|x+a \sin(x/a)|$. For the tangent to be parallel to the $x$-axis,the derivative $\frac{dy}{dx}$ must be $0$. Considering the case $y^2 = 4a(x + a \sin(x/a))$,we differentiate with respect to $x$: $2y \frac{dy}{dx} = 4a(1 + \cos(x/a))$. Setting $\frac{dy}{dx} = 0$,we get $1 + \cos(x/a) = 0$,which implies $\cos(x/a) = -1$. This condition implies $\sin(x/a) = 0$. Substituting $\sin(x/a) = 0$ back into the original equation,we obtain $y^2 = 4a(x + 0) = 4ax$. Thus,all such points lie on the parabola $y^2 = 4ax$.

All the points on the curve $y^{2}=4a|x+a \sin(x/a)|$,where the tangent is parallel to the $x$-axis,lie on:

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