Question

Let a variable line passing through the centre of the circle \(x^{2}+y^{2}-16 x-4 y=0\), meet the positive co-ordinate 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

Answer 1

Correct option is (2) 18

\((y-2)=m(x-8)\)

\(\Rightarrow \) x-intercept

\(\Rightarrow\left(\frac{-2}{\mathrm{m}}+8\right)\)

\(\Rightarrow \) 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