# $x$ and $y$ are two correlated variables with the same standard deviation and the correlation coefficient $r$ , the correlation coefficient between $x$ and $x+y$ is

$\begin {array} {1 1} (A)\;\large\frac{r}{2} & \quad (B)\;\sqrt{\large\frac{1-r}{2}} \\ (C)\;\sqrt{\large\frac{1+r}{2}} & \quad (D)\;0 \end {array}$

Let $u=x+y$
$\sigma =$ SD of $x\: or \: y$
$\therefore \sigma_u^2= \sigma_x^2+\sigma_y^2+2\: cov\: (x,y)$
$= \sigma^2+\sigma^2 2r+ \sigma^2$
$= 2\sigma^2(1+r)$
cov $(u,r) = E \{ ( u - \overline u)\: (x-\overline x) \}$
$= E \{ [ (x-\overline x)+(y-\overline y)] (x-\overline x) \}$
$= E \{ (x-\overline x)^2+(x-\overline x)(y-\overline y) \}$
$= \sigma^2(1+r)$
Thus $r(x, x+y)=\large\frac{\sigma^2(1+r)}{\sigma.\sigma \sqrt{2(1+r)}}$
$= \large\frac{\sqrt{r+1}}{2}$
Hence Ans : (C)