I have the following optimization problem I am unsure whether I've got it correct:
$$ \text{min } x^Tx \\ \text{so that } a + c^Tx <= 0 $$
I have introduced a slack variable $s$ to make it conceptually (for me) simpler:
$$ \text{min } x^Tx \\ \text{so that } a + c^Tx - s^2 = 0 $$
wich leades to the langragian $L = x^Tx + \lambda(a + c^Tx - s^2)$, from which I follow $2x = - \lambda c$ and from $a + c^Tx - s^2 = 0 \iff a + \lambda c^Tc - s^2 = 0$ that if $a >= 0$ then $\lambda = 0$ else $\lambda = -\frac{a}{<c,c>}$ (since we want to minimize $x^Tx = \lambda^2(c^Tc)$).
The stationary conditions are
$$ \cases{ 2x+\lambda c=0\\ a+c'x+s^2=0\\ \lambda s = 0 } $$
now developing, and considering the case $s=0$ we follow with
$$ \cases{ 2x+\lambda c=0\\ a+c'x=0 }\ \ \Rightarrow 2c'x+\lambda c'c=0\Rightarrow x = -\frac{a}{c'c}c $$
and calculating $\|x\|^2=\frac{a^2}{c'c}$. Now considering $\lambda=0$ we can calculate also the corresponding stationary point etc.