find the equation of the circle passing through the following 3 points (3,4) (3,2) and (1,4)
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Find the equation of the circle passing through the points (3,4),(3,2) and (1,4)
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ANSWER
Let the equation of circle be-
x
2
+y
2
+2gx+2fy+c=0.....(1)
Given that the circle passes through the points (3,4),(3,2) and (1,4).
Therefore,
(3)
2
+(4)
2
+2g(3)+2f(4)+c=0
⇒9+16+6g+8f+c=0
⇒6g+8f+c+25=0.....(2)
(3)
2
+(2)
2
+2g(3)+2f(1)+c=0
⇒9+4+6g+2f+c=0
⇒6g+2f+c+13=0.....(3)
(1)
2
+(4)
2
+2g(1)+2f(4)+c=0
⇒1+16+2g+8f+c=0
⇒2g+8f+c+17=0.....(4)
Subtracting equation (3) from (2), we have
(6g+8f+c+25)−(6g+2f+c+13)=0
6g+8f+c+25−6g−2f−c−13=0
⇒f=
6
−12
=−2
Now subtracting equation (4) from (2), we have
(6g+8f+c+25)−(2g+8f+c+17)=0
6g+8f+c+25−2g−8f−c−17=0
⇒g=
6
−8
=
3
−4
Now substituting the value of g and f in equation (2), we have
6(
3
−4
)+8(−2)+c+25=0
−8−16+c+25=0
⇒c=−1
Substituting the values of g,f and c in equation (1),
we get
x
2
+y
2
+2(
3
−4
)x+2(−2)y+(−1)=0
x
2
+y
2
−
3
8
x−4y−1=0
Hence the equation of circle passing through the given points is x
2
+y
2
−
3
8
x−4y−1=0.