JEE MAIN - Mathematics (2024 - 31st January Evening Shift - No. 18)
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 points $$A$$ and $$B$$. Then the minimum value of $$O A+O B$$, where $$O$$ is the origin, is equal to
12
20
24
18
Explanation
$$\begin{aligned}
& (y-2)=m(x-8) \\
& \Rightarrow x \text {-intercept } \\
& \Rightarrow\left(\frac{-2}{m}+8\right) \\
& \Rightarrow y \text {-intercept } \\
& \Rightarrow(-8 \mathrm{~m}+2) \\
& \Rightarrow \mathrm{OA}+\mathrm{OB}=\frac{-2}{\mathrm{~m}}+8-8 \mathrm{~m}+2 \\
& \mathrm{f}^{\prime}(\mathrm{m})=\frac{2}{\mathrm{~m}^2}-8=0 \\
& \Rightarrow \mathrm{m}^2=\frac{1}{4} \\
& \Rightarrow \mathrm{m}=\frac{-1}{2} \\
& \Rightarrow \mathrm{f}\left(\frac{-1}{2}\right)=18 \\
& \Rightarrow \text { Minimum }=18
\end{aligned}$$
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