$\mathrm{P}(-2 \sqrt{6}, \sqrt{3})$ lies on hyperbola

$\mathrm{e}=\frac{\sqrt{5}}{2} \Rightarrow \mathrm{b}^{2}=\mathrm{a}^{2}\left(\frac{5}{4}-1\right) \Rightarrow 4 \mathrm{~b}^{2}=\mathrm{a}^{2}$

$\text { Put in (i) } \Rightarrow \frac{6}{\mathrm{~b}^{2}}-\frac{3}{\mathrm{~b}^{2}}=1 \Rightarrow \mathrm{b}=\sqrt{3}$

$\Rightarrow \mathrm{a}=\sqrt{12}$

$\frac{x^{2}}{12}-\frac{y^{2}}{3}=1$ $[Image]$

Tangent at $\mathrm{P}$ :

$\frac{-x}{\sqrt{6}}-\frac{y}{\sqrt{3}}=1 \Rightarrow Q(0, \sqrt{3})$

Slope of $\mathrm{T}=-\frac{1}{\sqrt{2}}$

Normal at $P :$

$y-\sqrt{3}=\sqrt{2}(x+2 \sqrt{6})$

$\Rightarrow \quad R=(0,5 \sqrt{3})$

$Q R=6 \sqrt{3}$