Solve for the roots in the following equations

Question

I have two equalities: $$ \alpha x^{2} + \alpha y^{2} - y = 0 $$ $$ \beta x^{2} + \beta y^{2} - x = 0 $$

Where $$ \alpha, \beta $$ are both known constants.

How can I solve for $x$ and $y$?

Answer 1

The two curves are circles through the origin and they will intersect in at most one other point.

By eliminating $x^2+y^2$, we have

$$\beta y=\alpha x,$$

then

$$\beta(\beta x^2+\beta y^2-x)=(\beta^2 + \alpha^2)x^2 -\beta x = 0. $$

The rest is immediate.

$$\left(\frac\beta{\beta^2+\alpha^2},\frac\alpha{\beta^2+\alpha^2}\right).$$

We also have the special case $\alpha=\beta=0$, which is trivial.

Answer 2

The number of solutions depends on how many of $\alpha$ and $\beta$ are zero.

• For any $\alpha$ and $\beta$, $x = y = 0$ is a (trivial) solution.
• $\alpha = \beta = 0$: Only the trivial solution occurs.
• $\alpha \neq 0$ or $\beta \neq 0$ (or both): A nontrivial solution also appears, $x = \frac{\beta}{\alpha^2 + \beta^2}$ and $y = \frac{\alpha}{\alpha^2 + \beta^2}$.