Let a variable line passing through the centre of the circle x ^ 2...

Given: A line passing through the centre of circle ${x}^{2} + {y}^{2} - 16x - 4y = 0$ which is $\left(8, 2\right)$ intersects positive $x \text{\text{ and } }$ y axis at $A \text{\text{ and } }$ B respectively,

So, let the line be $\dfrac{x}{a} + \dfrac{y}{b} = 1$

Now, given line passes through the centre,

So, $\dfrac{8}{a} + \dfrac{2}{b} = 1$

Now, using $A . M \ge H . M$ we get,

$\dfrac{\dfrac{a}{2} + \dfrac{a}{2} + \dfrac{a}{2} + \dfrac{a}{2} + \dfrac{b}{1} + \dfrac{b}{1}}{6} \ge \dfrac{6}{\dfrac{2}{a} + \dfrac{2}{a} + \dfrac{2}{a} + \dfrac{2}{a} + \dfrac{1}{b} + \dfrac{1}{b}}$

$⇒ \dfrac{2a + 2b}{6} \ge \dfrac{6}{1}$

$⇒ 2a + 2b \ge 36$

$⇒ a + b \ge 18$

$⇒ OA + OB \ge 18$

Hence, the minimum value of $OA + OB = 18$