Equation of circle with center (0,3) and radius 2 isx2+(y−3)2=4Let 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+βα2+(β−3)2=β2+4+4β⇒α2=10(β−21)∴ Locus is x2=10(y−21) which is parabola.
