Question

Find all points of intersections of the 2 circles defined by the equations 
(x - 2)2 + (y - 2)2 = 4(x - 1)2 + (y - 1)2 = 4​

Answer 1

x2 - 4x + 2 + y2 - 4y + 2 = 4 : expand equation of first circle

x2 - 4x + 2 + y2 - 4y + 2 = 4 : expand equation of first circle x2 - 2x + 1 + y2 - 2y + 1 = 4 : expand equation of second circle

x2 - 4x + 2 + y2 - 4y + 2 = 4 : expand equation of first circle x2 - 2x + 1 + y2 - 2y + 1 = 4 : expand equation of second circle -2x - 2y - 6 = 0 : subtract the left and right terms of the above equations

x2 - 4x + 2 + y2 - 4y + 2 = 4 : expand equation of first circle x2 - 2x + 1 + y2 - 2y + 1 = 4 : expand equation of second circle -2x - 2y - 6 = 0 : subtract the left and right terms of the above equations y = 3 - x : solve the above for y.

x2 - 4x + 2 + y2 - 4y + 2 = 4 : expand equation of first circle x2 - 2x + 1 + y2 - 2y + 1 = 4 : expand equation of second circle -2x - 2y - 6 = 0 : subtract the left and right terms of the above equations y = 3 - x : solve the above for y. 2x2 - 6x + 1 = 0 : substitute y by 3 - x in the first equation, expand and group like terms.

x2 - 4x + 2 + y2 - 4y + 2 = 4 : expand equation of first circle x2 - 2x + 1 + y2 - 2y + 1 = 4 : expand equation of second circle -2x - 2y - 6 = 0 : subtract the left and right terms of the above equations y = 3 - x : solve the above for y. 2x2 - 6x + 1 = 0 : substitute y by 3 - x in the first equation, expand and group like terms. (3/2 + √(7)/2 , 3/2 - √(7)/2) , (3/2 - √(7)/2 , 3/2 + √(7)/2) : solve the above for x and use y = 3 - x to find y.

Answer 2

${\huge{\red{\mathfrak{answer}}}}$

x2 - 4x + 2 + y2 - 4y + 2 = 4 : expand equation of first circle x2 - 2x + 1 + y2 - 2y + 1 = 4 : expand equation of second circle -2x - 2y - 6 = 0 : subtract the left and right terms of the above equations y = 3 - x : solve the above for y. 2x2 - 6x + 1 = 0 : substitute y by 3 - x in the first equation, expand and group like terms. (3/2 + √(7)/2 , 3/2 - √(7)/2) , (3/2 - √(7)/2 , 3/2 + √(7)/2) : solve the above for x and use y = 3 - x to find y.

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