SOLUTION: If the point P(x,y) is equidistant from the points A(a+b,b-a) and B(a-b,a+b). Prove that bx=ay.

Algebra ->  Length-and-distance -> SOLUTION: If the point P(x,y) is equidistant from the points A(a+b,b-a) and B(a-b,a+b). Prove that bx=ay.      Log On


   

Question 1088304: If the point P(x,y) is equidistant from the points A(a+b,b-a) and B(a-b,a+b). Prove that bx=ay.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
D%5BPA%5D%5E2=%28x-%28a%2Bb%29%29%5E2%2B%28y-%28b-a%29%29%5E2
D%5BPA%5D%5E2=x%5E2-%282a%2B2b%29x%2Ba%5E2%2Bb%5E2%2By%5E2%2B%282a-2b%29y%2Ba%5E2%2Bb%5E2
D%5BPA%5D%5E2=x%5E2%2By%5E2-%282a%2B2b%29x%2B%282a-2b%29y%2B2a%5E2%2B2b%5E2
and similarly,
D%5BPB%5D%5E2=%28x-%28a-b%29%29%5E2%2B%28y-%28b%2Ba%29%29%5E2
D%5BPB%5D%5E2=x%5E2%2By%5E2%2B%282b-2a%29x-%282a%2B2b%29y%2B2a%5E2%2B2b%5E2
So then,
D%5BPA%5D%5E2=D%5BPB%5D%5E2

-%28a%2Bb%29x%2B%28a-b%29y=%28b-a%29x-%28a%2Bb%29y
-ax-bx%2Bay-by=bx-ax-ay-by
-bx%2Bay=bx-ay
2ay=2bx
highlight%28ay=bx%29