A curve having the condition that the slope of the tangent at some point is two times the slope of the straight line joining the same point to the origin of coordinates is a / an

parabola

Let $P(x, y)$ be any point on the curve. Then,

$\frac{d y}{d x}=2\left(\frac{y-0}{x-0}\right)$

$\Rightarrow \frac{1}{y} d y =\frac{2}{x} d x$

$\Rightarrow \log y=2 \log x+\log C \Rightarrow y=C x^2$         .....(i)

Clearly, it is a parabola.