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Q.

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

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a

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

b

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

c

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

d

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

answer is A.

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Detailed Solution

$x_{1} - a = x_{2} - x_{1} = . . . . . . . . = p$

$\begin{array}{l} \Rightarrow x_{1} = a + p, x_{2} = a + 2 p, .... . . . x_{n} = a + n p \\ h = a + \frac{(n + 1) p}{2} \\ \Rightarrow \frac{h - a}{p} = \frac{n + 1}{2} Similarly \frac{k - b}{q} = \frac{n + 1}{2} \\ \Rightarrow \frac{h - a}{p} = \frac{k - b}{q} \end{array}$

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