Explanation

(a)
Let the equation of line be $y = mx + c$. Since, this is the tangent to the circle$\ x^{2} + y^{2} = 5$
$\therefore\ \ c = \pm a\sqrt{1 + m^{2}}$
$= \pm \sqrt{5}\sqrt{1 + m^{2}}$ …(i)
Also, the above line is tangent to the parabola $y^{2} = 40x$
$\therefore\ c = \frac{a}{m} = \frac{10}{m}$
From Eqs.(i) and (ii), we get
$\frac{10}{m} = \pm \sqrt{5}\sqrt{1 + m^{2}}$
$\Rightarrow \ \ \ m^{4} + m^{2} - 20 = 0$
$\Rightarrow \ {(m}^{2} + 5)\left( m^{2} - 4 \right) = 0$
${\Rightarrow \ m}^{2} = 4,\ m^{2} \neq - 5$
$\Rightarrow \ m = \pm 2$
$\Rightarrow \ c = \pm 5$
$\therefore\ y = \pm 2x \pm 5$