If $a,x_1,x_2\mathrm{.....}$  and $b,y_1,y_2\mathrm{.....}$ form two Infinite A.P’s with common difference P and q respectively then the locus of $p\left(h,k\right)$ when

$h=\frac{x_1+x_2+x_3+\mathrm{......}+x_n}{n}$

$k=\frac{y_1+y_2+y_3+\mathrm{......}+y_n}{n}$ is

$h=\frac{x_1+x_2+x_3+\mathrm{....}+x_n}{n}$

$=\frac{a+p+a+2p+a+3p+---+a+np}{n}$

$=\frac{na+p.\frac{n\left(n+1\right)}{2}}{n}$

$=a+p\left(\frac{n+1}{2}\right)$

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

$k=\frac{y_1+y_2+y_3+\mathrm{......}+yn}{n}$

$=\frac{b+q+b+2q+b+3q+---+b+nq}{n}$

$=\frac{nb+q.\frac{n\left(n+1\right)}{2}}{n}$

$=b+q\left(\frac{n+1}{2}\right)$

$\frac{k-b}{q}=\frac{n+1}{2}$-----(2)$$

From (1) and (2) 

$\frac{h-a}{p}=\frac{k-b}{q}$

$q\left(h-a\right)=p\left(k-b\right)$

Locus of  is  $\left(h,k\right)$ is $q\left(x-a\right)=p\left(y-b\right)$