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Obtain the equation of the circle orthog...

Obtain the equation of the circle orthogonal to both the circles `x^2+y^2+3x-5y+6=0` and `4x^@+4y^2-28x+29=0` and whose centre lies on the line `3x+4y+1=0`

A

`4x^(2)+4y^(2)+2y-29=0`

B

`4x^(2)+4y^(2)+6y+5=0`

C

`2x^(2)+2y^(2)+3x+7y=0`

D

`x^(2)+y^(2)+3x-7y+3=0`

Text Solution

Verified by Experts

The correct Answer is:
A
Let equation of circle be `S-=x^(2)+y^(2)-2gx+2fy+c=0`
Condition of orthogonally `2g_(1)g_(2)+2f_(1)f_(2)=c_(1)+c_(2)`
Circle S and `S_(1)` are orthogonally
`:." "2g(3/2)+2f(-5/2)=c+6`
`3g-5f-c=6`………..(i)
Circles S and `S_(2)` are orthogonal
`:." "2g(-7/2)+2f(0)=c+29/4`
`-7g-c=29/4`...........(ii)
(i)-(ii) `10g-5f=-5/4`...........(iii)
Centre of `S(-g,-f)` lie on the line `3x+4y+1=0`
`:." "-3g-4f+1=0`..........(iv)
Solve equation (iii) and (iv)
`g=0,f=1/4,c=-29/4`
`:." "S=x^(2)+y^(2)+1/2y-29/4=0`
`4x^(2)+4y^(2)+2y-29=0`
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