The equation of the tangent to the hyperbola $\frac{{\mathrm{x}}^2}{4}-\frac{{\mathrm{y}}^2}{3}=1$ which is perpendicular to $\mathrm{x} + 2\mathrm{y} = 0$ is

for given hyperbola ${\mathrm{a}}^2=4, {\mathrm{b}}^2=3$.

slope of given line$\frac{-1}{2}$

Now slope of req. tangent $\left(\mathrm{m}\right)=2$ 

$\therefore$Equation of  req.tangent is 

$$\begin{gathered} \mathrm{y}=\mathrm{mx}\pm \sqrt{{\mathrm{a}}^2{\mathrm{m}}^2-{\mathrm{b}}^2} \\ \Rightarrow \mathrm{y}=2\mathrm{x}\pm \sqrt{13} \end{gathered}$$