#### Answer

a. $\sqrt{133}$
b. $(2\displaystyle \sqrt{2},\frac{3\sqrt{5}}{2})$

#### 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.
$P(3\sqrt{2},4\sqrt{5}), Q(\sqrt{2},-\sqrt{5})$.
$d(P, Q)=\sqrt{(\sqrt{2}-3\sqrt{2})^{2}+(-\sqrt{5}-4\sqrt{5})^{2}}$
$=\sqrt{(-2\sqrt{2})^{2}+(-5\sqrt{5})^{2}}=\sqrt{8+125}=\sqrt{133}$
b.
$M=(\displaystyle \frac{3\sqrt{2}+\sqrt{2}}{2},\frac{4\sqrt{5}+(-\sqrt{5})}{2})=(2\sqrt{2},\frac{3\sqrt{5}}{2})$