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If $a, x_{1}, x_{2}, x_{3} ....$ and $b, y_{1}, y_{2}, .....$ form two infinite A.P’s with common difference p and q respectively then the locus of $P (h, k)$ when
$h = \frac{x_{1} + x_{2} + x_{3} ....... + x_{n}}{n}$ , $k = \frac{y_{1} + y_{2} + ..... + y_{n}}{n}$ is

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

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

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

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

answer is A.

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Detailed Solution

$x_{1} - a = x_{2} - x_{1} =____= p$ ; $x_{1} = a + p$

$x_{n} = a + n p;$ $\frac{Σ x_{i}}{n} = a + \frac{(n + 1) p}{2}$

$x = a + \frac{(n + 1) p}{2}$ ; $\frac{x - a}{p} = \frac{n + 1}{2}$

similarly $\frac{y - b}{q} = \frac{n + 1}{2}$ ; $\frac{x - a}{p} = \frac{y - b}{q}$

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

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