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 $\mathrm{A}$ and $\mathrm{B}$. Then the minimum value of $\mathrm{OA}+\mathrm{OB}$, where $\mathrm{O}$ 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
$\Rightarrow(-8 \mathrm{~m}+2)$
$\Rightarrow \mathrm{OA}+\mathrm{OB}=\frac{-2}{\mathrm{~m}}+8-8 \mathrm{~m}+2$
$\mathrm{f}^{\prime}(\mathrm{m})=\frac{2}{\mathrm{~m}^{2}}-8=0$
$\Rightarrow \mathrm{m}^{2}=\frac{1}{4}$
$\Rightarrow \mathrm{m}=\frac{-1}{2}$
$\Rightarrow \mathrm{f}\left(\frac{-1}{2}\right)=18$
$\Rightarrow$ Minimum $=18$
