If $sx+ty=1$ for some $0<s,t<1$ and $x,y\in\mathbb{C}$, can we conclude that $x,y\geq0$?

Question

If $sx+ty=1$ for some $0<s,t<1$ (with $s+t=1$, but not sure if this is needed) and $x,y\in\mathbb{C}$, can we conclude that $x,y\geq0$ (i.e. $x,y\in\mathbb{R}$ and $x,y\geq0$)? This may be a really elementary question, but I failed to work out the details. I tried to visualize $\mathbb{C}$ as a real 2-dimensional real vector space (where $1\cong(1,0)$ and $i\cong(0,1))$. Then the above yields $(1,0)=(sx_{1}+ty_{1},sx_{2}+ty_{2})$, which proves that $sx_{1}+ty_{1}=0$, i.e. the imaginary part of $sx+ty$ vanishes. Am I thinking in the right direction? Any help would be greatly appreciated!

Answer 1

The statement is not true, as $s=t=0.5,x=1+i,y=1-i$ is a counterexample to it :)

Answer 2

For $s>0$ we have $t=1-s$ and $y=\frac{sx-1}{s-1}$ which is valid for any $x\in\mathbb{C}$.