Problem:
Let L be the line with slope 125β that contains the point A=(24,β1), 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 (β14,27) in the original system has coordinates (Ξ±,Ξ²) in the new coordinate system. Find Ξ±+Ξ².
Solution:
The equations for L and M are 5xβ12yβ132=0 and 12x+5yβ90=0, respectively. Because P lies in the second quadrant in the new coordinate system, it follows that
Ξ±= negative distance from P to M=122+52βββ£12β (β14)+5β 27β90β£β=13β123β, and
Ξ²= positive distance from P to L=122+52ββ£5β (β14)β12β 27β132β£β=13526β.