Circle Question 10
Question 10 - 2024 (31 Jan Shift 2)
Let a variable line passing through the centre of the circle $x^{2}+y^{2}-16 x-4 y=0$, meet the positive coordinate axes at the point $A$ and $B$. Then the minimum value of $OA+OB$, where $O$ is the origin, is equal to
(1) 12
(2) 18
(3) 20
(4) 24
Show Answer
Answer (2)
Solution
$(y-2)=m(x-8)$
$\Rightarrow x$-intercept
$\Rightarrow\left(\frac{-2}{m}+8\right)$
$\Rightarrow y$-intercept
$\Rightarrow(-8 m+2)$
$\Rightarrow OA+OB=\frac{-2}{m}+8-8 m+2$
$f^{\prime}(m)=\frac{2}{m^{2}}-8=0$
$\Rightarrow m^{2}=\frac{1}{4}$
$\Rightarrow m=\frac{-1}{2}$
$\Rightarrow f\left(\frac{-1}{2}\right)=18$
$\Rightarrow$ Minimum $=18$
