Equation of circle with center (0,3) and radius 2 is x2+(y−3)2=4 Let locus of the variable circle is (α,β) As it touches x-axis. ∴ It's equation is (x−α)2+(y+β)2=β2
Circle touch externally ⇒c1c2=r1+r2 ∴√α2+(β−3)2=2+β α2+(β−3)2=β2+4+4β ⇒α2=10(β−1∕2) ∴ Locus is x2=10(y−