First we will need to expand both equations:
(x-2)**2 + (y-2)**2 =4 expands to x**2-4x+4 + y**2-4y +4 = 4 which simplifies to x**2 - 4x + y**2-4y + 8 = 4
(x-1)**2 + (y-1)**2 = 4 expands to x**2-2x+1 + y**2-2y+1 = 4 which simplifies to x**2 - 2x +y**2-2y + 2 = 4
Now we'll subtract one of these equations from the other:
x**2 - 4x + y**2 -4y + 8 = 4
- (x**2 - 2x + y**2 -2y + 2 = 4)
which gives us
-2x -2y +6 = 0.
Now we will solve for one of the variables and plug it back into one of the original equations to solve for the other variable. Let's choose to solve for x:
-2x - 2y = -6
x + y = 3 (dividing both sides by -2)
x = 3-y
Plug that value of x for x in the first of the two expanded equations:
x**2 - 4x + y**2 -4y + 8 = 4
(3-y)**2 -4(3-y) + y**2 - 4y + 8 = 4 (plug in 3-y for x)
9-6y+y**2 -12 + 4y + y**2 -4y + 8 = 4 (expand)
2y**2 - 6y + 5 = 4 (combine like terms)
2y**2 + 6y + 1 = 0
Using the quadratic formula where a=2, b=6 and c=1, we get:
x=-0.177 and 2.82
Our difference equation was x = 3 - y --> so y = 3 - x
Plugging the above x values into that equation we will get the corresponding y values:
y = 3 - (-0.177) = 3.177, so the first coordinate of intersection is (-0.177,3.177)
y = 3 - (2.82) = 0.18, so the second coordinate of intersection is (2.82, 0.18)
