Problem:
Let L be the line with slope 125​ that contains the point A=(24,−1), and let M be the line perpendicular to line L that contains the point B=(5,6). The original coordinate axes are erased, and line L is made the x-axis and line M the y-axis. In the new coordinate system, point A is on the positive x-axis, and point B is on the positive y-axis. The point P with coordinates (−14,27) in the original system has coordinates (α,β) in the new coordinate system. Find α+β.
Solution:
The equations for L and M are 5x−12y−132=0 and 12x+5y−90=0, respectively. Because P lies in the second quadrant in the new coordinate system, it follows that
α= negative distance from P to M=122+52​−∣12⋅(−14)+5⋅27−90∣​=13−123​, and
β= positive distance from P to L=122+52​∣5⋅(−14)−12⋅27−132∣​=13526​.