Let the maximum and minimum values of (√8x-x^2-12-4)^2+(x-7)^2,x R be M and m, respectivel…

Question

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

Answer: 1600

Accepted Answer

$\begin{gathered} y=\sqrt{8x-{x}^{2}-12}\Rightarrow (x-4{)}^{2}+{y}^{2}={2}^{2} \ \Rightarrow d=(y-4{)}^{2}+(x-7{)}^{2} \end{gathered}$

$\begin{gathered} \Rightarrow M=P{A}^{2}=16+25=41 \ m=P{Q}^{2}=(\sqrt{16+9}-2{)}^{2}=9 \ \Rightarrow {M}^{2}-{m}^{2}=1681-81=1600 \end{gathered}$

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