A particle moves in 1-dimension. If we plot its velocity and displacement over time, the trajectory forms a circle that's centered at the origin.

Which of the following relations is true regarding its acceleration $(a),$ velocity $(v),$ and displacement $(x)$ ?

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Note:
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In the options,
$k$
is a positive constant.

$a= kx$
$a = -k x$
$a = kv$
$a = - kv$

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$v^2+x^2=C^2$

Differentiating with respect to $x$ ,

$2v\frac{dv}{dx}+2x=0 \\ a=-x$