The tangent and normal to the ellipse $3{\mathrm{x}}^2+5{\mathrm{y}}^2=32$ at the point $\mathrm{P}(2, 2)$ meet the x-axis at $\mathrm{Q}$ and $\mathrm{R},$ respectively. Then the area (in sq.units) of the triangle $\mathrm{PQR}$ is:

$\begin{array}{rcl}\frac{\mathrm{dy}}{\mathrm{dx}}& =& \frac{-6\mathrm{x}}{10\mathrm{y}}=\frac{-3\mathrm{y}}{5\mathrm{y}}\\ {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_{(2, 2)}& =& \frac{-3}{5}\end{array}$

eq of Tangent $3\mathrm{x}+5\mathrm{y}=16$

eq of normal   $5\mathrm{x}-3\mathrm{y}=4$

meet x-axis $\Rightarrow \mathrm{Q} \left(\frac{16}{3}, 0\right) \mathrm{R} \left(\frac{4}{5}, 0\right)$

$\mathrm{P}(2, 2) \mathrm{Q}\left(\frac{16}{3}, 0\right) \mathrm{R}\left(\frac{4}{5}, 0\right)$

area $∆ \mathrm{PQR}=\frac{1}{2}\left|\begin{array}{cc}\frac{-10}{3}& \frac{6}{5}\\ 2& 2\end{array}\right|$

                        $=\frac{1}{2}\left|\frac{-20}{3} \frac{-12}{5}\right|=\frac{136}{2\times 5}=\frac{68}{15}$