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.

Answer 1

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)