Solutions to a polynomial equation with constraint

Question

I am looking for solutions to the following simple polynomial equation, $$ x_1^2 + x_2^2 + y_1^2 + y_2^2 = -2 (x_1 x_2 + 3 y_1 y_2) $$ where $x_i, y_i \in \mathbb{R}$. Importantly, I would like only solutions for which $y_1, y_2 >0$.

Thus far, I have been unable to find any solutions satisfying this later constraint, despite the fact that there seems to be no simple proof against their existence.

Is there any easy way to find such solutions?

Answer 1

Hint.

$$ (x_1+x_2)^2+(y_1+y_2)^2 = -4y_1y_2 $$

now if $y_i > 0$...