The random variables $X$ and $Y$ are related by the equation $aX+bY+c = 0$.We have to show that the correlation between them is $-1$ if the signs of $a$ and $b$ are the same and $+1$ if they are different.
Intuitively I thought of this $Y = (-\frac{a}{b}) X + (-\frac{c}{b})$ and if the slope is positive then the correlation is 1 and if the slope is negative then the correlation is -1 which yields as per the result that - the correlation between them is $-1$ if the signs of $a$ and $b$ are the same and $+1$ if they are different.
But how to do it more formally, also I am curious to see that in the answer the value of $\rho$ is independent of $c$.
Yes I also tried $\rho = \frac{E(XY) -E(X)E(Y)}{\sigma_{X}\sigma_{Y}}$
But it looks a bit lengthy, is there any other procedure to solve this ?
What's problematic about using your expression for $\rho$? Insert $Y = (-\frac{a}{b}) X + (-\frac{c}{b})$ and use the linearity of $E$ and almost-linearity of $\sigma$ (I don't know whether that property has a proper name) to get an expression with $E(X),E(X^2)$ and $\sigma_X$. From there it simplifies nicely.