The equation of the circle, orthological to both the circle `x^(2) + y^(2) + 3x - 5y + 6 = 0` and `x^(2) + 4y^(2) - 28x + 29 = 0` and whose centre lies on the line `3x + 4y + 1 = 0` is

Options

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`

Answer

A

Explanation

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`