If a circle Passes through a point (1,2) and cut the circle `x^2+y^2 = 4` orthogonally,Then the locus of its centre is

Options

A. `x^(2)+y^(2)-3x-8y+1=0`
B. `x^(2)+y^(2)-2x-6y-7=0`
C. `2x+4y-9=0`
D. `2x+4y-1=0`

Answer

C

Explanation

Let the circle be `x^(2)+y^(2)+2gx+2fy+c=0`. This passes through (1, 2).
`:. 5+2g+4f+c=0` ...(i)
The circles `x^(2)+y^(2)=4 and x^(2)+y^(2)+2gx+2fy+c=0` cut orthogonally.
`:. 2(gxx0+fxx0c-4 rArr c=4`
Putting c=4 (i), we get 2g+4f+9=0.
Therefore, the locus of (-g, -f) is
`-2x-4y+9=0 or, 2x+4y-9=0`.