find the equation of circle having centre in 1st quadrant, touching x-axis,having a common tangent y=3^1/2x+4 with the circle x^2+y^2+4x+4y+4=0 such that the distance between two circles along the x-axis is 3 units ?

Answer 1

Is that y = (√3)x+4 ?

The other circle is clearly

(x+2)^2 + (y+2)^2 = 4

and the way the question is worded, it appears that the first circle is also tangent to the x-axis at (1,0), making it

(x-1)^2 + (y-k)^2 = k^2

Now it should not be too hard to find k so that the desired line is a common tangent.