Calculate $\mathbb{E}(X-Y\mid 2X+Y).$ if $X\sim N(0,a)$ and $Y\sim N(0,b)$ 
Question: Given that $X$ and $Y$ are two random variables  satisfying $X\sim N(0,a)$ and $Y\sim N(0,b)$ for some $a,b>0$. Assume that $X$ and $Y$ have correlation $\rho.$ 
  Calculate 
  $$\mathbb{E}(X-Y \mid 2X+Y).$$

I tried to use the fact that if $A$ and $B$ are independent, then $\mathbb{E}(A\mid B) = \mathbb{E}(A)$ and uncorrelated implies independence in jointly normal distribution. 
So, I attempted to express $X-Y$ as a linear combination of $2X+Y$ and $Z$ where $\operatorname{Cov}(2X+Y,Z) = 0.$
But I am not able to do so.
Any hint is appreciated.

By @Kavi Rama Murthy answer (and me  in other answer)
$$E(X-Y|2X+Y)=A(2X+Y)$$ 
Now By the Projection property ,$E(X-Y|2X+Y)$ minimized 
$$E(X-Y-g(2X+Y))^2$$ conditional-expectation-as-best-predictor
I want to find $A$ by minimizing $E(X-Y-A(2X+Y))^2$
$$E(X-Y-A(2X+Y))^2=E((1-2A)X-(1+A)Y)^2$$
$$=E((1-2A)X)^2+E((1+A)Y)^2
-2E((1-2A)X (1+A)Y)2$$
$$=(1-2A)^2E(X)^2+(1+A)^2E(Y)^2
-2(1-2A)(1+A)E(X Y)$$
$$=(1-2A)^2a+(1+A)^2b
-2(1-2A)(1+A)cou(X Y)$$
$$=(1-2A)^2a+(1+A)^2b
-2(1-2A)(1+A)\rho \sqrt{a}\sqrt{b}$$
by derivation $\frac{d}{dA}$ and equal to $0$
$$\frac{d}{dA} E((1-2A)X-(1+A)Y)^2=0$$
$$\Leftrightarrow$$
$$0=
-4(1-2A)a+2(1+A)b-2(-2)(1+A)\rho \sqrt{a}\sqrt{b}-2(1-2A)\rho \sqrt{a}\sqrt{b}$$
$$\Leftrightarrow$$
$$0=\bigg(
-4a+2b+4\rho \sqrt{a}\sqrt{b}-2\rho\sqrt{a}\sqrt{b}
\bigg)+\bigg(
8a+2b+4\rho \sqrt{a}\sqrt{b}+4\rho \sqrt{a}\sqrt{b}
\bigg)A$$
$$\Leftrightarrow$$
$$0=\bigg(
-4a+2b+2\rho \sqrt{a}\sqrt{b}
\bigg)+\bigg(
8a+2b+8\rho \sqrt{a}\sqrt{b}
\bigg)A$$
$$\Leftrightarrow$$
$$A=\frac{2a-b-\rho \sqrt{a}\sqrt{b}}{4a+b+4\rho \sqrt{a}\sqrt{b}}$$
