What is the product of all the coordinates of all the points of intersection of the two circles defined by x^2-2x+y^2-10y+20=0 and x^2-8x+y^2-10y+35=0?
+191 Add and to the first equation and and to the second equation to find that the given equations are equivalent to
\begin{align*}
(x^2-2x+1)+(y^2-10y+25)&=1\text{, and} \\
(x^2-8x+16)+(y^2-10y+25)&=4
\end{align*}
which are equivalent to
\begin{align*}
(x-1)^2+(y-5)^2 &=1^2, \\
(x-4)^2+(y-5)^2 &=2^2,
\end{align*}
The two circles have centers (1,5) and (4,5) and radii 1 and 2. Since the centers of the circles are units apart and the sum of their radii is , the two circles intersect at only one point. We can see that (2,5) is the desired intersection point, so our product is 2*5= 10
