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Question
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.
Solution
P(x, y) is equidistant from the points A(a + b, b – a) and B(a – b, a + b).
∴ AP = BP
∴ `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
= x2-2x(a-b)+(a-b)2+y2-2y(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 ...(proved)
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