Minimum value of $ab+bc+ca$ depending on given constraints

You're doing it right. Another way of looking at what you're doing: You have $$ab + bc + ac = {1 \over 2} (a + b + c)^2 - {1 \over 2}(a^2 + b^2 + c^2)$$ Under the constraint that $a^2 + b^2 + c^2$, you therefore have $$ab + bc + ac = {1 \over 2} (a + b + c)^2 - {1 \over 2}$$ You're trying to minimize this quantity. Since ${1 \over 2} (a + b + c)^2 \geq 0$, it's smallest when $a + b + c = 0$, in which case you have $$ab + bc + ac = -{1 \over 2}$$ This is the minimum possible value of $ab + bc + ac$, and it is achieved for any $(a,b,c)$ for which $a^2 + b^2 + c^2 = 1$ and $a + b + c = 0$. There are many such $(a,b,c)$ as they correspond to the intersection of the sphere $a^2 + b^2 + c^2 = 1$ and the plane through the origin with equation $a + b + c = 0$.