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Question
If the point (x, y) is equidistant from the points (a + b, b – a) and (a – b, a + b), prove that bx = ay.
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Solution
Let P(x, y), Q(a + b, b – a) and R (a – b, a + b) be the given points. Then
PQ = PR ...(Given)
⇒ `sqrt({x - (a + b)}^2 + {y - (b - a)}^2) = sqrt({x - (a - b)}^2 + {y - (a + b)}^2`
⇒ `{x - (a + b)}^2 + {y - (b - a)}^2 = {x - (a - b)}^2 + {y - (a + b)}^2`
⇒ x2 – 2x(a + b) + (a + b)2 + y2 – 2y(b – a) + (b – a)2 = x2 + (a – b)2 – 2x(a – b) + y2 – 2(a + b) + (a + b)2
⇒ –2x(a + b) – 2y(b – a) = –2x(a – b) – 2y(a + b)
⇒ ax + bx + by – ay = ax – bx + ay + by
⇒ 2bx = 2ay
⇒ bx = ay
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