Question 752632: Determine the equation of the circle circumscribing the triangle determined by the lines x+y=8, 2x+y=14 and 3x+y=22
Answer by dkppathak(439)   (Show Source):
You can put this solution on YOUR website! triangle determined by lines are x+y=8, 2x+y=14 and 3x+y=22
solving these equations in two groups by elimination
x+y=8, 2x+y=14 we will get x=6 and y=2 so vertex coordinates are (6 2)
similarly 2x+y=14 and 3x+y=22 we will get vertex as (8 -2)
x+y=8, 3x+y=22 we will get vertex as (7 1)
therefore the circle will pass through vertex of triangle
circle passing through (6 2) (8 -2) and (7 1)
we know that general equation of circle
x^2+y^2+2gx+2fy+c=0
circle passing through(6 2) will be
12g+4f+c= -40 (i)
circle passing through (8 -2) will be
16g-4f+c=-68 (ii)
circle passing through (7 1) we will get
14g +2f +c =-50 (iii)
by solving equations in two groups
12g+4f+c= -40 (i)
16g-4f+c=-68
we will get
-4g+8f =28
same way using two equation
16g-4f+c=-68
14g +2f +c =-50(iv)
we will get
2g-6f=-18 (v)
by solving equation (iv) and (v)
we will get
f=2 and g=-3
by putting value of g and f in equation (iii)
we will get c=-12
arranging these value of g=-3 f=2 and c=-12 in general equations of circle
equation of circle will be
x^2 +y^2-6x+4y-12 =0
ANSWER equation of circle subscribe the triangle will be
x^2 +y^2-6x+4y-12 =0
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