A circles 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)

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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