Circles are drawn through the point (3,0) to cut an intercept of length 6 units on the negative direction of the x-axis. The equation of the locus of their centres is

Options

A. y=0
B. y=x
C. x=0
D. y=-x

Answer

C

Explanation

Let the equation of the circle be
`x^(2)+y^(2)+2gx+2fy+c=0 " " ...(i)`
If passes through (3, 0) . Therefore,
`9+6g+c=0 " " ...(ii)`
Circle (i) cuts an intercept of 6 units on the negative direction of x-axis.
`:. 2sqrt(g^(2)-c)=6 rArr g^(2)-c=9 " " ...(iii)`
From (ii) and (iii), we get
`9+6g+g^(2)=9`
`rArr g(g+6)=0rArr g = 0 " " [:' g gt 0 :. g + 6 !=0]`
Hence, the locus of (-g, -f) is - x = 0, i.e. x = 0.