# A circles touches the $\dpi{100} x-$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.

P Plabita

As we learnt in

Circle touching x-axis and having radius r -

$x^{2}+y^{2}\pm 2rx+2fy+f^{2}= 0$

- wherein

Where f is a variable parameter.

General form of a circle -

$x^{2}+y^{2}+2gx+2fy+c= 0$

- wherein

centre = $\left ( -g,-f \right )$

radius = $\sqrt{g^{2}+f^{2}-c}$

Let centre of circle be P(h,k)

distance  AB =$\sqrt{h^{2}+\left ( k-3 \right )^{2}}$

$\sqrt{h^{2}+\left ( k-3 \right )^{2}}=\left ( k+2 \right )$

Squaring both sides

$h^{2}+\left ( k-3 \right )^{2}=\left ( k+2 \right )^{2}$

$h^{2}+k^{2}-6k+9=k^{2}+4+4k$

$h^{2}=10k-5$

Which represents a parabola.

Option 1)

a circle

Incorrect option

Option 2)

an ellipse

Incorrect option

Option 3)

a parabola

Correct option

Option 4)

a hyperbola.

Incorrect option

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