Answer
$(5,0),(0,5)$
Work Step by Step
The line $x+y=5$ intersects the circle $x^2+y^2=25$ at $(5,0)\,$and$\,(0,5)$ as can be seen from the figure.
The solution can also be obtained analytically by solving the system of equations formed by the two equations $x+y=5\,\,$ and$\,\,x^2+y^2=25$
$$\because x+y = 5 \,\,\,\,\,\, \therefore x = 5-y\\ $$
Substituting for ($x = 5-y$) in ($x^2+y^2=25$)
$$(5-y)^2+y^2 = 25\\(25-10y+y^2)+y^2=25\\2y^2-10y=0\\2y(y-5) = 0$$
$\therefore y = 0$ and $ y = 5$
$x|_{y=0} = 5-0 = 5$
$x|_{y=5} = 5-5 = 0$
$\therefore$ The coordinates of the points of intersection are $(5,0)$ and $(0,5)$.
