#### Answer

a. $3\sqrt{55}$
b. $(2\displaystyle \sqrt{7},\frac{7\sqrt{3}}{2})$

#### Work Step by Step

Let $P(x_{1}, y_{1})$ and $Q(x_{2}, y_{2})$.
$d(P, Q)=\sqrt{(x_{2}-x_{\mathrm{I}})^{2}+(y_{2}-y_{1})^{2}}$
$M=(\displaystyle \frac{x_{\mathrm{I}}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$
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a.
$P(-\sqrt{7},8\sqrt{3}), Q(5\sqrt{7},-\sqrt{3})$.
$d(P, Q)=\sqrt{(5\sqrt{7}-(-\sqrt{7}))^{2}+(-\sqrt{3}-8\sqrt{3})^{2}}$
$=\sqrt{(6\sqrt{7})^{2}+(-9\sqrt{3})^{2}}=\sqrt{36(7)+81(3)}$
$=\sqrt{9(4\cdot 7+9\cdot 3)}=3\sqrt{28+27}=3\sqrt{55}$
b.
$M=(\displaystyle \frac{-\sqrt{7}+5\sqrt{7}}{2},\frac{8\sqrt{3}+(-\sqrt{3})}{2})=(2\sqrt{7},\frac{7\sqrt{3}}{2})$