# If two variables $x, y$ are related by $ax+by+c=0$ with $ab < 0$ then correlation coefficient between $x$ and $y$ is given by

$\begin {array} {1 1} (A)\;1 & \quad (B)\;-1 \\ (C)\;0 & \quad (D)\;None\: of \: these \end {array}$

$ax+by+c=0$
$\Rightarrow a\overline x + b \overline y +c=0$
$\Rightarrow a(x- \overline x)+b(y- \overline y)=0$
$x- \overline x= \large\frac{-b}{a} (y- \overline y)$
Now $cov (x,y)= \large\frac{1}{n} \Sigma (x- \overline x) \: (y- \overline y)$
$= -\large\frac{b}{a} \: \large\frac{1}{n} \Sigma (y-\overline y)^2$
$-\large\frac{b}{a} \sigma_y^2$
Also $\sigma_x^2 = \large\frac{1}{n} \Sigma (x - \overline x)^2$
$= \large\frac{b^2}{a^2} \bigg( \large\frac{1}{n} \Sigma ( y - \overline y ) ^2 \bigg)$
$= \large\frac{b^2}{a^2} \sigma_y^2$
$\therefore r = \large\frac{cov (x,y)}{\sigma_x \sigma_y}$
$= \large\frac{-\large\frac{b}{a} \sigma_y^2}{\sqrt{\large\frac{b^2}{a^2} \sigma_y^2. \sigma_y}}$
$= \large\frac{-\large\frac{b}{a}}{|\large\frac{b}{a}|}$ $= \large\frac{-\large\frac{b}{a}}{-\large\frac{b}{a}}$
$= 1$
Ans : (A)