The centres of those circles which touch the circle, x2+y2−8x−8y−4=0, externally and also touch the x-axis, lie on :

 

  • Option 1)

     a circle.

     

  • Option 2)

    an ellipse which is not a circle.

     

  • Option 3)

     a hyperbola.

     

  • Option 4)

     a parabola.

 

Answers (2)

As we learnt in

Common tangents of two circles -

When two circles touch  each other externally, there are three common tangents, two of them are direct.

 

- wherein

 

 

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.

 

 

Standard equation of parabola -

x^{2}=4ay

- wherein

 

 Circle: x^{2}+y^{2}-8x-8y-4=0

externally

radius of circle touching x-axis=k,

we get C_{1}C_{2}=\sqrt{(h-4)^{2}+(k-4)^{2}}=k+6

On squaring both sides

h^{2}+k^{2}-8h-4k+32=k^{2}+36+12k

h^{2}=8h+16=16k+20

\left ( h-4 \right )^2=16\left ( k+\frac{5}{4} \right )

compared to x^{2}=4AY

Represent a parabola


Option 1)

 a circle.

 

Incorrect Option

Option 2)

an ellipse which is not a circle.

 

Incorrect Option

Option 3)

 a hyperbola.

 

Incorrect Option

Option 4)

 a parabola.

Correct option

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