The locus of a point whose distance from the y-axis is half of its distance from origin is

let $\mathrm{p}({\mathrm{x}}_1, {\mathrm{y}}_1)$ be locus of point. 

Given that, distance form P to y-axis $=\frac{1}{2}$ (distance from P to origin)

$$\begin{gathered} \Rightarrow \left|{\mathrm{x}}_1\right|=\frac{1}{2}\left(\sqrt{{{\mathrm{x}}_1}^2+{{\mathrm{y}}_1}^2}\right) \\ \Rightarrow 2\left|{\mathrm{x}}_1\right|=\sqrt{{{\mathrm{x}}_1}^2+{{\mathrm{y}}_1}^2} \\ \Rightarrow 4{{\mathrm{x}}_1}^2={{\mathrm{x}}_1}^2+{{\mathrm{y}}_1}^2 \\ \Rightarrow {{\mathrm{y}}_1}^2=3{{\mathrm{x}}_1}^2 \\ \therefore \mathrm{locus} \mathrm{of} \mathrm{P} \mathrm{is} {\mathrm{y}}^2=3{\mathrm{x}}^2 \end{gathered}$$