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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. The locus of $p (α, β),$ wher $α = \frac{x_{1} + x_{2} + . .. + x_{n}}{n}$ and $p = \frac{y_{1} + y_{2} + . .. + y_{n}}{n}$ is

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

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

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

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

answer is D.

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Detailed Solution

$α = \frac{x_{1} + x_{2} + . ..... x_{n}}{n} = \frac{\frac{n}{2} [2 (p + a) + (n - 1) a]}{n} = p + a + \frac{(n - 1) a}{2} \Rightarrow α - p = \frac{a (n + 1)}{2} \to (1)$ $similarly β - q = b (n + 1) / 2 \to (2)$

$\frac{(1)}{(2)} \Rightarrow \frac{α - p}{β - q} = \frac{a}{b} \Rightarrow b (α - p) = a (β - q)$
Locus of P is $b (x - p) = a (y - q)$

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