Assume that for some $a\geq 0$, $b \geq b_0 > 0$ and $c \geq c_0 >0$ we have the following: $$a^2+b^2 \leq c^2$$ Can I conclude that $$a \leq c - \alpha b$$ for some $\alpha > 0$? If yes, what is $\alpha$? Note that both $b$ and $c$ are lower bounded by some known positive constant.
I know it does not hold in general for $\alpha \geq 1$ and it obliviously holds for $\alpha = 0$, but I wonder if it hold for some other positive $\alpha$ perhaps as a function of $b_0$ and $c_0$.
No, such an estimate does not exist. For $b_0 > 0$, $c_0 > 0$ and $x > 0$ choose $$ b = b_0 \,, \, c = c_0 + x \,, \, a = \sqrt{c^2 - b^2} \, . $$ Then $a^2+b^2=c^2$ and $$ \frac{c-a}{b} = \frac{b}{c+a} \le \frac{b}{c} = \frac{b_0}{c_0 + x} $$ and that becomes arbitrarily small for $x \to \infty$.