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Question

# Obtain the equation of the circle orthogonal to both the circles x2+y2+3x−5y+6=0 and 4x2+4y2−28x+29=0 and whose centre lies on the line 3x+4y+1=0

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Solution

## Given circles are x2+y2+3x−5y+6=0.................(i)and 4x2+4y2−28x+29=0or x2+y2−7x+294=0..............(ii)Let the required circle be x2+y2+2gx+2fy+c=0....................(iii)Since circle (iii) cuts circles (i) and (ii) orthogonally∴2g(32)+2f(−52)=c+6 or 3g−5f=c+6 ..............(iv)and 2g(−72)+2f.0=c+294or−7g=c+294.............(v)From (iv) & (v) we get 10g−5f=−54 or 40g - 20f = -5 ..............(vi)Given line is 3x+4y=−1 .............(vii)Since centre (-g, -f) of circle (iii) lies on line (vii) .............(viii)Solving (vi) & (viii) we get g=0,f=14 ∴from(5),c=−294∴ From (iii) required circle is x2+y2+12y−294=0 or 4(x2+y2)+2y−29=0

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