Find equation of circle having centre in first quadrant, touching x axis, having common tangent y=sqrt(3)x +4with circle x^2 + y^2 + 4x + 4y + 4 = 0 such that distance b/w two circles along x axis is 3 units.

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Your question is wrong.We haveC:x2+y2+4x+4y+4=0Centre is (−2,2) and radius is (−2)2+(−2)2−4−−−−−−−−−−−−−−−√=4+4−4−−−−−−−√=2If y=3√x+4 touches C then perpendicular from centre to line must beequal to radius of circle.y=3√x+4 ⇒3√x−y+4=0Length of perpendicular=∣∣3√×(−2)−2+4∣∣(3√)2+(−1)2√=∣∣23√+2∣∣2=2(3√+1)2=3√+1 which is not equal to radius of cirlcle. Hence line cannot be tangentto cirlce.Hence information given in question is incorrect.

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