When $|a-b|=|a|-|b|$?

Question

Could someonoe help me to decide if the following satetement is true?

If $K$ is a strictly convex Banach space and $a,b\in{K}$ verify $|a-b|=|a|-|b|=1$ then, $a=\lambda{b}$ for some $\lambda\geq{0}$.

I only need the case $K=\mathbb{R}$ and $K=\mathbb{C}$ but I wonder if there exists a proof in a general strictly convex banach space.

Thanks.