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Prove that the equation x^(2)+y^(2)-2x-2...

Prove that the equation `x^(2)+y^(2)-2x-2ay-8=0, a in R ` represents the family of circles passing through two fixed points on x-axis.

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We have `x^(2)+y^(2)-2x-2ay-8=0`.
`:. (x^(2)+y^(2)-2x-8)-2a(y)=0,a in R`
This equation is of the form `S+ lambda L=0`, which represents family of circle passing through points of intersection of circle `x^(2)+y^(2)-2x-8=0` and line `y=0` (x-axis).
Solving circle and line, we have
`x^(2)-2x-8=0`
`implies (x-4)(x+2)=0`
`implies x=4,x= -2`
So, circle pass through two fixed points (4,0) and (-2,0) on x-axis.
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