Find the coordinates of the point of intersection of the line $x + y =2$ with the circle $x^2 + y^2 = 4$.
Equation of given circle: $\;\;\;$ $x^2 + y^2 = 4$ $\;\;\; \cdots \; (1)$
Equation of given line: $\;\;\;$ $x + y = 2$ $\implies$ $y = 2 - x$ $\;\;\; \cdots \; (2)$
Substitute the value of $y$ from equation $(2)$ in equation $(1)$. We have
$x^2 + \left(2 - x\right)^2 = 4$
i.e. $x^2 - 4x = 0$
i.e. $x \left(x - 4\right) = 0$
i.e. $x = 0$ $\;$ or $\;$ $x = 4$
$\therefore$ $\;$ We have from equation $(2)$,
when $x = 0$, $y = 2 - 0 = 2$ and
when $x = 4$, $y = 2 - 4 = -2$
$\therefore$ $\;$ The required coordinates of the point of intersection of the line $x + y = 2$ with the circle $x^2 + y^2 = 4$ are $\left(0,2\right)$ and $\left(4,-2\right)$.