Answer: \(y=4x-30\); \(AB=\dfrac{15\sqrt{17}}{2}\).

(a) The gradient of the line joining \((12,1)\) and \((4,3)\) is

\(\frac{3-1}{4-12}=\frac2{-8}=-\frac14.\)

The perpendicular gradient is therefore

\(4.\)

The midpoint of the two points is

\(\left(\frac{12+4}{2},\frac{1+3}{2}\right)=(8,2).\)

So the perpendicular bisector has equation

\(y-2=4(x-8).\)

Hence

\(y=4x-30.\)

(b) The \(y\)-intercept is found by putting \(x=0\):

\(y=-30.\)

The \(x\)-intercept is found by putting \(y=0\):

\(0=4x-30,\)

so

\(x=\frac{15}{2}.\)

The length \(AB\) is therefore

\(AB=\sqrt{\left(\frac{15}{2}\right)^2+30^2}.\)

Thus

\(AB=\sqrt{\frac{225}{4}+900} =\sqrt{\frac{3825}{4}} =\frac{15\sqrt{17}}{2}.\)