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Distance between Two Points on the Same Coordinate Axes

If the point ...

Question

If the point P(p, q) is equidistant from the points A(a+b, b-a) and B(a-b, a+b), then

ap = bq

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bp + aq = 0

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ap + bq = 0

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- bp + aq = 0

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Solution

The correct option is D
- bp + aq = 0

Given that AP = BP
$∴ A P^{2} = B P^{2} \Rightarrow (a + b - p)^{2} + (b - a - q)^{2} = (a - b - p)^{2} + (a + b - q)^{2} \Rightarrow a^{2} + b^{2} + p^{2} + 2 a b - 2 b p - 2 a p + b^{2} + a^{2} + q^{2} - 2 a b + 2 a q - 2 b q = a^{2} + b^{2} + p^{2} - 2 a b + 2 b p - 2 a p + a^{2} + b^{2} + q^{2} + 2 a b - 2 b q - 2 a q$

$\Rightarrow 2 a b - 2 b p - 2 a p - 2 a b + 2 a q - 2 b q = - 2 a b + 2 b p - 2 a p + 2 a b - 2 b q - 2 a q$

$\Rightarrow - 4 b p + 4 q a = 0 \Rightarrow - b p + a q = 0$

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