y=¼x+2 and 4y-x=-60.
Rewrite these as 4y-x=2 and 4y-x=-60. It’s clear the lines are parallel.
A common perpendicular is given by negating and inverting the slope. The slope is 1/4 so the slope of the perpendicular is -4. The general equation of the perpendicular is y=-4x+c where c is a constant.
If we pick a point on one of the parallel lines we can make the perpendicular pass through the point. Let’s pick (0,2) on the first line, so this must also be on the perpendicular: 2=c so the perpendicular has the equation y=2-4x. The perpendicular meets the second line when 4(2-4x)-x=-60.
8-17x=-60, 17x=68, x=4 so 4y=x-60=-56 and y=-14.
We have two points now (0,2) and (4,-14). Using Pythagoras we can find the length of the line joining the points:
√(4²+(-14-2)²)=√(16+256)=√272=4√17=16.4924 approx.