Practice problem

Substitution in Systems of Nonlinear Equations

Let $a,b$ be real numbers. Which of the following statements are always true?

Choose all correct answers

If $ax^2+by^2 = 0$ has a real solution $(x,y)\ne (0,0)$ , then $ax^2+by^2 = 1$ has a real solution $(x,y)$ .

If $ax^2+by^2 = 1$ has a real solution $(x,y)$ , then $ax^2+by^2 = 0$ has a real solution $(x,y)\ne (0,0)$ .

If $ax^2+by^2 = z^2$ has a real solution $(x,y,z)\ne (0,0,0)$ , then $ax^2+by^2 = 1$ has a real solution $(x,y)$ .

If $ax^2+by^2 = 1$ has a real solution $(x,y)$ , then $ax^2+by^2 = z^2$ has a real solution $(x,y,z)\ne (0,0,0)$ .