Not Simultaneously Zero

Question

I am reviewing Chapter 4, Section 6.1, Theorem 6 of D. Widder's Advanced Calculus. It states

$1. f(x,y,\alpha),g(\alpha),h(\alpha) \in C^1$

$2. f_1^2+f_2^2 \ne 0$

$3. (g')^2+(h')^2 \ne 0$

$4. f(g(\alpha),h(\alpha),\alpha) \equiv 0$

$5. f_3(g(\alpha),h(\alpha),\alpha)) \equiv 0$

$\implies$ The family $f(x,y,\alpha)=0$ has the curve $x=g(\alpha),y=h(\alpha)$ as an envelope.

Questions

1. Does (2) above mean that $f_1$ and $f_2$ can't be simultaneously zero? (i.e. $f_1$ and $f_2$ must be smooth functions)

2. Same question as 1 for (3) above.

3. Why do we sum-square? (e.g. why $f_1^2+f_2^2 \ne 0$ instead of $f_1+f_2 \ne 0$?)

Answer 1

1) Yes, assuming we are talking about real-valued functions, $f_1^2 + f_2^2 \ne 0$ if and only if $f_1$ and $f_2$ are not both $0$.

2) Same question, same answer.

3) $f_1 + f_2$ would be $0$ if $f_2 = -f_1$, where both can be nonzero.