Question: I've started with thinking that if X and Y are perfectly correlated, then it is the same as looking at the correlation of Y and -Y (since the X provides no new information) and thus the correlation is -1.

Is this correct?

Answer: 
Hint: *In general*, \begin{align}
\rho\_{A,B} &= \frac{\operatorname{cov}(A,B)}{\sqrt{\operatorname{var}(A)\operatorname{var}(B)}},\\
\operatorname{var}(X\pm Y)&= \operatorname{var}(X)+\operatorname{var}(Y)
\pm 2\operatorname{cov}(X,Y),\\
\text{and}\qquad\operatorname{cov}(X+Y,X-Y)&=\operatorname{var}(X)-\operatorname{var}(Y)\end{align} So, work out what $\rho\_{X+Y,X-Y}$ is in general, and in the special case when $Y = aX+b$. You might be surprised at the result.
If $X$ is a linear function of $Y$ (definition of perfect correlation), then both $X-Y$ and $X+Y$ will be linear functions of $Y$, and therefore are linear functions of each other. 

So, $X-Y$ and $X+Y$ are perfectly correlated.