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  • 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 isOption: 1 a circl

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

Option: 1

a circle 


Option: 2

an ellipse 


Option: 3

a parabola 


Option: 4

a hyperbola 


Answers (1)

best_answer

Let locus of centre be α, β, then according to given condition, if \mathrm{r_1, r_2} are radii of circles, then

Internal touch. This case does not exist as centre of circle is (0, 3) and radius is 2.

\mathrm{\begin{aligned} & \quad C_1 C_2=r_2 \pm r_1 \Rightarrow \sqrt{(\alpha-0)^2+(\beta-3)^2}=|\beta \pm 2| \\ & \Rightarrow \quad \alpha^2+\beta^2-6 \beta+9=\beta^2+4+4 \beta \\ & \text { and } \alpha^2+\beta^2-6 \beta+9=\beta^2-4 \beta+4 \\ & \Rightarrow \quad \alpha^2-10 \beta+5=0 \text { and } \alpha^2=2 \beta-5 \\ & \Rightarrow \quad x^2=10 y-5 \text { and } x^2=2 y-5 \end{aligned}}

\mathrm{\text { Both are parabolas but } x^2=2 y-5 \text { does not exist. }}

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Anam Khan

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