Let the lengths of intercepts on x -axis and y -axis made by the...

${x}^{2} + {y}^{2} + ax + 2ay + c = 0$

$2\sqrt{{g}^{2} - c} = 2\sqrt{\dfrac{{a}^{2}}{4} - c} = 2\sqrt{2}$

$⇒ \dfrac{{a}^{2}}{4} - c = 2 . . . \left(1\right)$

$2\sqrt{{f}^{2} - c} = 2\sqrt{{a}^{2} - c} = 2\sqrt{5}$

$⇒ {a}^{2} - c = 5 . . . \left(2\right)$

\left(1\right) \text{ and } \left(2\right)$</p><p>$\dfrac{3{a}^{2}}{4} = 3 ⇒ a = - 2\left(a < 0\right)$</p><p>$∴ c = - 1$</p><p>Circle$⇒ {x}^{2} + {y}^{2} - 2x - 4y - 1 = 0$</p><p>$⇒ {\left(x - 1\right)}^{2} + {\left(y - 2\right)}^{2} = 6$</p><p>Given$x + 2y = 0 ⇒ m = - \dfrac{1}{2}$</p><p>${m}_{tangent} = 2$</p><p>Equation of tangent$⇒ \left(y - 2\right) = 2\left(x - 1\right) \pm \sqrt{6}\sqrt{1 + 4}$</p><p>$⇒ 2x - y \pm \sqrt{30} = 0$</p><p>Perpendicular distance from \left(0, 0\right) = \left|\dfrac{\pm \sqrt{30}}{\sqrt{4 + 1}}\right| = \sqrt{6}$