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Question

# A circle touches the x-axis and also touches the circle with center at (0,3) and radius2. The locus of the center of the circle is

A
an ellipse
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B
a circle
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C
a hyperbola
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D
a parabola
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Solution

## The correct option is D a parabolaEquation of circle with centre (0, 3) and radius 2 is, x2+(y−3)2=4 Let locus of the variable circle is (α,β) It touches x-axis. It equation (x−α)2+(y−β)2=β2 Given circles 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−1/2), which is parabola.

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