Equation of the hyperbola with one focus at the origin and directrix x + 3 = 0 and eccentricity $\sqrt{3}$ is

$$\begin{gathered} Given focus=(0,0) and e=\sqrt{3} \\ Given directrix is x+3=0 \\ Let P =(x,y) be any point on hyperbola then \\ SP=ePM \\ \Rightarrow S{P}^2=e^{2}P{M}^2 \end{gathered}$$

 $$\begin{gathered} \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2=3\frac{(\mathrm{x}+3{)}^2}{1} \\ \Rightarrow 3x^2+18x+27-x^2-y^2=0 \\ \Rightarrow 2x^2-y^2+18x+27=0 \end{gathered}$$