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Question

# The equation of the circle which cuts orthogonally each of the three circles given below: x2+y2−2x+3y−7=0, x2+y2+5x−5y+9=0 and x2+y2+7x−9y+29=0

A
x2+y216x8y12=0
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B
x2+y216x18y4=0
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C
x2+y2+16x18y+4=0
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D
x2+y2+16x+18y+12=0
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Solution

## The correct option is B x2+y2−16x−18y−4=0Let the required circle be x2+y2+2gx+2fy+c=0 ⋯(1) Since, it is orthogonal to three given circles respectively, therefore 2g×(−1)+2f×32=c−7 or −2g+3f=c−7……(2) 2g×52+2f×(−52)=c+9 or 5g−5f=c+9……(3) and 2g×72+2f×(−92)=c+29 or 7g−9f=c+29……(4) Solving equations (2),(3) and (4) we get g=−8,f=−9 and c=−4 Substituting the values of g, f, c in (1) then required circle is x2+y2−16x−18y−4=0 Alternate Solution: S1:x2+y2−2x+3y−7=0S2:x2+y2+5x−5y+9=0S3:x2+y2+7x−9y+29=0 radical axis for S1 and S2 S2−S1=0⇒7x−8y+16=0⋯(1) radical axis for S2 and S3 S3−S2=0⇒x−2y+10=0⋯(2) From (1) and (2) radical centre of the given circle will be (8,9) Hence equation of the required circle will be (x−8)2+(y−9)2=(√(S1))2⇒x2+y2−16x−18y−4=0

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