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in Coordinate Geometry by (55.4k points)
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If the point (x, y) be equidistant from the points (a + b, b - a) and (a - b, a + b), prove that (a - b)/(a + b) = (x - y)/(x + y).

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Given points are A(a + b, b - a) and B(a - b, a + b). It is told that S(x, y) is equidistant from A and B.

So, we get SA = SB,

We know that distance between two points (x1, y1) and (x2, y2) is

Now,

⇒ SA = SB

⇒ SA2 = SB2

⇒ (x - (a + b))2 + (y - (b - a))2 = (x - (a - b))2 + (y - (a + b))2

⇒ x2 - 2(a + b)x + (a + b)2 + y2 - 2(b - a)y + (b - a)2 = x2 - 2(a - b)x + (a - b)2 + y2 - 2(a + b)y + (a + b)2

⇒ x(- 2a - 2b + 2a - 2b) = y(2b - 2a - 2a - 2b)

⇒ x(- 4b) = y(- 4a)

⇒ x(b) = y(a)

⇒ x/y = a/b

Applying componendo and dividendo,

∴ Thus proved.

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