If $x_2(t)/x_1(t)=y_2(t)/y_1(t)$, then is $\vec Y(t)=(y_1(t),y_2(t))$ a reparametrization of $\vec X(t)=(x_1(t),x_2(t))$?

Question

The definition of a parametrized curve $\vec X:I\rightarrow \mathbb{R}^2$ was given by a continuous function $g:J\rightarrow I$, then $\vec \xi =\vec X \circ g:J\rightarrow \mathbb{R}^2$ a reparametrization of $\vec X(t)$.

However, if $x_2(t)/x_1(t)=y_2(t)/y_1(t)$, then is $\vec Y(t)=(y_1(t),y_2(t))$ a reparametrization of $\vec X(t)=(x_1(t),x_2(t))$?