Find the equations of the tangents to the hyperbola x² – 4y² = 4 which are
i) parallel and
ii) perpendicular to the line x + 2y = 0

Given hyperbola $\frac{{\mathrm{x}}^2}{4}-\frac{{\mathrm{y}}^2}{1}=1\Rightarrow {\mathrm{a}}^2=4,{\mathrm{b}}^2=1$
i) Any line parallel to x + 2y = 0 is
 x + 2y +K=0 ...(1)
Since this is a tangent to hyperbola then
$\begin{array}{l}\Rightarrow {\mathrm{a}}^2{\mathrm{l}}^2-{\mathrm{b}}^2{\mathrm{m}}^2={\mathrm{n}}^2\Rightarrow 4\left(1\right)-1\left(4\right)={\mathrm{k}}^2\\ \Rightarrow 4\left(1\right)-1\left(4\right)={\mathrm{k}}^2\Rightarrow \mathrm{k}=0\end{array}$
$\therefore \text{ No parallel tangent to }\mathrm{x}+2\mathrm{y}=0$
ii) Any line perpendicular to $\mathrm{x}+2\mathrm{y}=0 is 2\mathrm{x}-\mathrm{y}+\mathrm{\lambda }=0 ...\left(2\right)$
Since this touches the hyperbola then
$\begin{array}{l}\Rightarrow {\mathrm{a}}^2{\mathrm{l}}^2-{\mathrm{b}}^2{\mathrm{m}}^2={\mathrm{n}}^2\\ \Rightarrow 4\left(4\right)-1\left(1\right)={\mathrm{\lambda }}^2\Rightarrow \mathrm{\lambda }=\pm \sqrt{15}\end{array}$
$\text{∴ Required tangents are }2\mathrm{x}-\mathrm{y}\pm \sqrt{15}=0$