$X$ and $Y$ are standardized r.v.s. Find $a,b,c,d$ such that $Z=aX+bY$ and $W=cX+dY$ are uncorrelated but still standardized.

Question

Let $X$ and $Y$ be standardized r.v.s (i.e., marginally they each have mean $0$ and variance $1$) with correlation $\rho \in (−1, 1)$. Find $a, b, c, d$ (in terms of $\rho$) such that $Z = aX + bY$ and $W = cX + dY$ are uncorrelated but still standardized.

Here is my attempt at the solution:

Since they are uncorrelated, then:

$$Corr(Z,W)=0=Corr(aX+bY,cX+dY)=\frac{Cov(aX+bY,cX+dY)}{SD(Z)(SD(W)}$$

This implies that $Cov(aX+bY,cX+dY)=0$, thus,

$$Cov(aX+bY,cX+dY)=Cov(aX,cX)+Cov(aX,dY)+Cov(bY,dY)+Cov(bY,cY) = acVar(X)+(ad+bc)Cov(X,Y)+bdVar(Y)=ac+(ad+bc)\rho+bd=0$$

Since they are both standardized,

$$Var(Z)=1=a^2+b^2+2ab\rho$$

and,

$$Var(W)=1=c^2+d^2+2cd\rho$$

So far, I have 3 equations but 4 unknowns and I am not sure how to get the 4th equation. Anyone have some insight into this question?