Q36 of 77 Page 7


If the distances of A(x, y) from P(a + b, b - a) and Q(a - b, a + b) are equal,prove that bx = ay.


SOLUTION:

AP 2 = (a + b - x) 2 + (b - a - y) 2
AQ 2 = (a - b - x) 2 + (a + b - y) 2
Since, AP = AQ (Given)
⇒ AP 2 = AQ 2
⇒ (a + b - x) 2 + (b - a - y) 2 = (a - b - x) 2 + (a + b - y) 2
⇒ a 2 + b 2 + x 2 + 2ab – 2ax – 2bx + b 2 + a 2 + y 2 – 2ba – 2by + 2ay = a 2 + b 2 + x 2 – ⇒ 2ab + 2bx – 2ax + a 2 + b 2 + y 2 + 2ab – 2ay – 2by – 2bx + 2ay = 2bx – 2ay
⇒ 4ay = 4bx
⇒ ay = bx
Hence proved.

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