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Question
If the point (x, y) is at equidistant from the point (a + b, b – a) and (a-b, a + b). Prove that ay = bx.
Solution
Given that (x, y) is equidistant from the point (a + b, b - a) and (a - b, a + b).
Hence, distance of (x, y) from both points will be same.
Hence, `sqrt((y - b + a)^2 + (x - a - b)^2)`
= `sqrt((y - a - b)^2 + (x - a + b)^2)`
On squaring and expanding :
y2 + b2 + a2 - 2by - 2ab + 2ay + x2 + a2 + b2 - 2ax + 2ab - 2bx
= y2 + a2 + b2 - 2ay + 2ab - 2by + x2 + a2 + b2 - 2ax - 2ab + 2bx
2ay - 2bx = 2bx - 2ay
4ay = 4bx
⇒ ay = bx
Hence proved.
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