Don't understand an algebraic answer

Question

Question:

$a+\frac{10b}{a^2+b^2} = 5$

$b+\frac{10a}{a^2+b^2} = 4$

Prove $ab$ is not equal to 0.

Thanks for your help.

Answer 1

Let $a=0$.

Thus, from the first equation we obtain $b=2$ and from the second we obtain $b=4$, which is a contradiction.

By the same way we can get a contradiction for $b=0$.

Thus, $ab\neq0.$

Answer 2

Multiplying both equations, rearranging and factoring gives $$ab\left(1+\frac {100}{(a^2+b^2)^2}\right)=10$$ hence $ab\neq 0$.