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"the points A,B,C are on the circle."the points A,B,C are on the circle.
"we can write the fallowing equations"we can write the fallowing equations
A(5,2)A(5,2)
x^2+y^2+ax+bx+c=0x2+y2+ax+bx+c=0
5^2+2^2+5a+2b+c=052+22+5a+2b+c=0
25+4+5a+2b+c=025+4+5a+2b+c=0
5a+2b+c=-29" (1)"5a+2b+c=−29 (1)
B(8,1)B(8,1)
x^2+y^2+ax+bx+c=0x2+y2+ax+bx+c=0
8^2+1^2+8a+b+c=082+12+8a+b+c=0
64+1+8a+b+c=064+1+8a+b+c=0
8a+b+c=-65" (2)"8a+b+c=−65 (2)
C(3,4)C(3,4)
x^2+y^2+ax+bx+c=0x2+y2+ax+bx+c=0
3^2+4^2+3a+4b+c=032+42+3a+4b+c=0
9+16+3a+4b+c=09+16+3a+4b+c=0
3a+4b+c=-25" (3)"3a+4b+c=−25 (3)
"let's add the equation (2) to (3)"let's add the equation (2) to (3)
11a+5b+2c=-90" (4)"11a+5b+2c=−90 (4)
"expand the equation (1) by 2"expand the equation (1) by 2
10a+4b+2c=-58" (5)"10a+4b+2c=−58 (5)
"subtract the equation (5) from (4)"subtract the equation (5) from (4)
a+b=-32" (5)"a+b=−32 (5)
"subtract the equation (1) from (2)"subtract the equation (1) from (2)
3a-b=-36" (6)"3a−b=−36 (6)
"now add (5) to (6)"now add (5) to (6)
4a=-684a=−68
color(red)(a=-17)a=−17
"use (5)"use (5)
-17+b=-32−17+b=−32
b=-32+17b=−32+17
color(red)(b=-15)b=−15
"use (2)"use (2)
8(-17)-15+c=-658(−17)−15+c=−65
-136-15+c=-65−136−15+c=−65
c=-65+151c=−65+151
color(red)(c=86)c=86
"r:radius of circle"r:radius of circle
r=sqrt(a^2+b^2-4c)/2r=√a2+b2−4c2
r=sqrt(289+225-344)/2r=√289+225−3442
r=sqrt(514-344)r=√514−344
r=sqrt(170)/2r=√1702
r^2=170/4=42.5r2=1704=42.5
area=pi*r^2area=π⋅r2
area=42.5piarea=42.5π
