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.
I'll treat this as [self-study](/questions/tagged/self-study "show questions tagged 'self-study'"), and I'd encourage you to read [its wiki](https://stats.stackexchange.com/tags/self-study/info) and add the tag.

Your argument is already very good. Here are a few pointers. Feel free to write a comment so we can discuss and work towards a good answer.

* I assume you are looking at Pearson's correlation, right? (Does your argument work for other measures of correlation?)
* What does a perfect correlation mean *graphically*?
* What will $X+Y$ and $X-Y$ look like *graphically* if $X$ and $Y$ are perfectly Pearson-correlated?