Let a variable line passing through the centre of the circle x 2 + y 2 − 16 x − 4 y = 0 , meet the positive…

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

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

Now, given line passes through the centre,

So, $\frac{8}{a}+\frac{2}{b}=1$

Now, using $A.M\ge H.M$ we get,

$\frac{{\displaystyle \frac{a}{2}+\frac{{\displaystyle a}}{{\displaystyle 2}}+\frac{{\displaystyle a}}{{\displaystyle 2}}+\frac{{\displaystyle a}}{{\displaystyle 2}}+\frac{{\displaystyle b}}{{\displaystyle 1}}+\frac{{\displaystyle b}}{{\displaystyle 1}}}}{6}\ge \frac{6}{\frac{2}{a}+\frac{{\displaystyle 2}}{{\displaystyle a}}+\frac{{\displaystyle 2}}{{\displaystyle a}}+\frac{{\displaystyle 2}}{{\displaystyle a}}+\frac{{\displaystyle 1}}{{\displaystyle b}}+\frac{{\displaystyle 1}}{{\displaystyle b}}}$

$\Rightarrow \frac{{\displaystyle 2a+2b}}{6}\ge \frac{6}{1}$

$\Rightarrow 2a+2b\ge 36$

$\Rightarrow a+b\ge 18$

$\Rightarrow OA+OB\ge 18$

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