Let the maximum and minimum values of (√8x - x^2 - 12 - 4)^2 + (x - 7)^2, x ∈ R

Question

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

Answer 1

Correct answer:1600

\((x-7)^{2}+(y-4)^{2}\)

\(y=\sqrt{8 x-x^{2}-12}\)

\(y^{2}=-(x-4)^{2}+16-12\)

\((x-4)^{2}+y^{2}=4\)

\(\mathrm{m}=9\)

\(\mathrm{M}=41\)

\(\mathrm{M}^{2}-\mathrm{m}^{2}=41^{2}-9^{2}=1600\)