Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length `2sqrt7` on y-axis is (are)

Options

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

Answer

A

Explanation

Here, length of intercept on Y-axis
`=2sqrt(f^(2)-c)`
and if circle touches X-axis
`impliesg^(2)=c`
for `x^(2)+y^(2)+2gx+2fy+c=0`
passes through (3,0) .

`implies9+6g+c=0` . . . (i)
`g^(2)=c` . . . (ii) ltbrnd `2sqrt(t^(2)-c)=2sqrt(7)`
`impliesf^(2)-c=7` . . . (iii)
From eqs. (i) and (ii),
`g^(2)+6g+9=0 implies(g+3)^(2)=0`
`thereforeg=-3 and c=9`
`thereforef^(2)=7+c=16`
`impliesf=+-4`
`thereforex^(2)+y^(2)-6x+-8y+9=0`