color(red)("There may be an easier way of doing this")
The trick is to manipulate these 3 equations in such a way that you end up with 1 equation with 1 unknown.
Consider the standard form of (x-a)^2+(y-b)^2=r^2
Let point 1 be P_1->(x_1,y_1) =(0,8)
Let point 2 be P_2->(x_2,y_2)=(5,3)
Let point 3 be P_3->(x_3,y_3)=(4,6)
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For P_1 ->(x_1-a)^2+(y_1-b)^2=r^2
(0-a)^2+(8-b)^2=r^2
a^2+64-16b+b^2=r^2...............Equation(1)
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For P_2->(x_2-a)^2+(y_2-b)^2=r^2
(5-a)^2+(3-b)^2=r^2
25-10a+a^2+9-6b+b^2=r^2
a^2-10a+34-6b+b^2=r^2............Equation(2)
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For P_3->(x_3-a)^2+(y-b)^2=r^2
(4-a)^2+(6-b)^2=r^2
16-8a+a^2+36-12b+b^2=r^2
a^2-8a+52-12b+b^2=r^2...........Equation(3)
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Lets see where this gets us!
Equation(3) - Equation(2)
a^2-8a-12b+b^2+52=r^2
ul(a^2-10a-6b+b^2+34=r^2) larr" subtract"
0" "+2a -6b+0+18=0
2a-6b+18=0 ...........................Equation(4)
=>a=(6b-18)/2 = 3b-9
color(brown)("we can now substitute for "a)color(brown)("in equations 1 and 2 and solve for "b)
equation(1)=r^2=equation(2)
a^2-16b+b^2" " =" " a^2-10a-6b+b^2+34
cancel(a^2)-16b+cancel(b^2)" " =" " cancel(a^2)-10a-6b+cancel(b^2)+34
Substituting for a
-16b" "=" "-10(3b-9)-6b+34
-16b" "=" "-30b+90-6b+34
-16b" "=" "-36b+124
" "color(green)(ul(bar(|" "b=124/20=31/5" "|))
color(red)("I will let you take it on from this point")
