Explanation

(a)
Equation of normal to the hyperbola at the point $(5\sec{\theta,\ 4\tan{\theta)}}$ is
$5x\cos{\theta + 4y\cot{\theta = 25 + 16}}$ …(i)
This line is perpendicular to the lie $2x + y = 1.$
$\therefore\ \ m_{1}m_{2} = - 1$
$\Rightarrow \left( \frac{- 5\cos\theta}{4\cot\theta} \right)( - 2) = - 1$
$\Rightarrow \ \sin{\theta = - \frac{2}{5}}$
$\therefore\cos{\theta = \sqrt{1 - \frac{4}{25}} = \mp \frac{\sqrt{21}}{5}}$
and $\cot{\theta = \mp \frac{\sqrt{21}}{2}}$
from Eq.(i)
$5x\frac{\sqrt{21}}{5} - \frac{4y\sqrt{21}}{2} = 41$
$\Rightarrow \ \sqrt{21}(x - 2y) = 41$