Answer
a) $4\sqrt 3$
(b) $(4\sqrt 3,3\sqrt 5)$
(c) $y=3\sqrt 5$
Work Step by Step
(a) Given $P(2\sqrt 3,3\sqrt 5), Q(6\sqrt 3,3\sqrt 5)$, we can fine the distance $d(P, Q)=\sqrt {(6\sqrt 3-2\sqrt 3)^2+(3\sqrt 5-3\sqrt 5)^2}=4\sqrt 3$
(b) the coordinates of the midpoint of the segment PQ can be found as $(\frac{6\sqrt 3+2\sqrt 3}{2},\frac{3\sqrt 5+3\sqrt 5}{2})$ or $(4\sqrt 3,3\sqrt 5)$
(c) Assume the equation passing the two points as $y=mx+b$, we have $m=\frac{3\sqrt 5-3\sqrt 5}{6\sqrt 3-2\sqrt 3}=0$ and $3\sqrt 5=0(2\sqrt 3)+b$ or $b=3\sqrt 5$, thus the equation is $y=3\sqrt 5$
