find the equation of circle which is orthogonal to each of the following circles x^2+y^2+4x+2y+4=0 2x^2+2y^2+8x+6y-3=0 x^2+y^2+6x-2y-3=0.
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The Circle having Center at the radical center of the Three given circles and radius equal to the length of the tangent from it to any one of the three circles cuts all three circles orthogonally
Given Circles are x
2
+y
2
+2x+4y+1=0..................(1)
2(x
2
+y
2
)+6x+8y−3=0..................(2)
x
2
+y
2
−2x+6y−3=0..................(3)
For circles S
1
=0,S
2
=0
the radical axes is given by S
1
−S
2
=0
The radical axes of (i),(ii),(iii) are respectively
x−
2
5
=0
10x−4y+3=0
From solving above Equations
x=
2
5
,y=7
Thus, the Coordinates of Radical Circle are
2
5
,7
Length of Tangent from (
2
5
,7) to (1)
r=
4
25
+49+5+28+1
=
2
357
Hence, Required Circle is (x−
2
5
)
2
+(y−7)
2
=
4
357
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