A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is ?

Hello student,

Equation 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
Circles touch externally
∴ 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.

Thanks and Regards

Shaik Aasif

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