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If $p, x_{1}, x_{2} x_{3} ....$ and $q, y_{1}, y_{2}, y_{3} .....$ form two infinite AP’s with common differences a and b respectively, then locus of $P (α, β)$ , where $α = \frac{x_{1} + x_{2} + ..... + x_{n}}{n}$ and $β = \frac{y_{1} + y_{2} + ..... + y_{n}}{n}$ is

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$a (x - p) = b (y - q)$

$p (x - a) = q (y - b)$

$p (x - p) = q (y - q)$

$b (x - p) = a (y - q)$

answer is D.

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Detailed Solution

$α = \frac{(p + a) + (p + 2 a) + ...... + (p + n a)}{n} = \frac{n p}{n} + \frac{a n (n + 1)}{2 n}$

$α - p = \frac{(n + 1) a}{2} ∴ \frac{α - p}{a} = \frac{n + 1}{2}$

Simillarly $\frac{β - q}{b} = \frac{n + 1}{2} .$

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