Question

if the point P (x, y) is equidistant from point A (a+b, b-a)& B (a-b , a+b). Prove that bx = by

Answer 1

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Answer 2

AP = BP

AP = √[ x - (a + b)]² + [y - (b - a)]²
       =  √(x - a - b)² + (y - b + a)² ..........(i)

BP = √[ x - (a - b)]² + [y - (a + b)]²
       = √( x - a + b)² + (y - a + b)²........(ii)

Equating (i) and (ii),
√(x - a - b)² + (y - b + a)² = √(x - a + b)² + (y - a + b)²
(x - a - b)² + (y - b + a) = (x - a + b) + (y - a + b)

x² + a² + b² - 2xa - 2ab + 2bx + y² + b² + a² - 2ya + 2ab - 2by = x² + a² + b² - 2xa + 2ab - 2bx + y² + a² + b² - 2yb - 2ab + 2ay

x² + 2a + 2b² - 2xa + 2bx + y² - 2ya - 2by = x² + 2a² + 2b² - 2xa - 2bx + y² - 2yb + 2ay

4bx = 4ay
bx = ay