# 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.

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

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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