Let the maximum and minimum values of $\left(\sqrt{8 x-x^2-12}-4\right)^2+(x-7)^2, x \in \mathbf{R}$ be $\mathrm{M}$ and $\mathrm{m}$, respectively. Then $\mathrm{M}^2-\mathrm{m}^2$ is equal to _________

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Aicharya · Accepted Answer

$\begin{aligned} & (x-7)^2+(y-4)^2 \ & y=\sqrt{8 x-x^2-12} \ & y^2=-(x-4)^2+16-12 \ & (x-4)^2+y^2=4\end{aligned}$ $\begin{aligned} & m=9 \ & M=41 \ & M^2-m^2=41^2-9^2=1600\end{aligned}$

Let the maximum and minimum values of $\left(\sqrt{8 x-x^2-12}-4\right)^2+(x-7)^2, x \in \mathbf{R}$ be…

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