Mathematics - Circle Question with Solution | TestHub

Given: A line passing through the centre of circle $x^2+y^2-16x-4y=0$ which is $\left(8,2\right)$ intersects positive $x \& y$ axis at $A \& B$ 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 $OA+OB=18$