Is the family of circles $x^2+y^2-((a^2+b^2+1+2fb)/a)x+2fy+1=0$ coaxial for constants $a$, $b$ and parameter $f$?

Question

The equation of a family of circles is -

$$x^2+y^2-((a^2+b^2+1+2fb)/a)x+2fy+1=0$$ where $a$, $b$ are real constants and $f$ is a real parameter.

How can I check whether this family of circles is coaxial or not?

I want to know how to approach the problem (i.e. what is the condition for being coaxial)

Answer 1

Converting a comment to an answer, as requested.

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A convenient thing about circle equations is that, if you combine two of them in such a way that the $x^2$ and $y^2$ terms cancel, then what remains is the equation of the radical axis.

So, consider two versions of your equation: one with parameter $f$ and one with parameter, say, $g\neq f$. Check whether the equation of the radical axis depends upon $f$ or $g$.

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Since OP has correctly solved the problem in the comments to the question, I'll include a little bit more.

In this case, it can be helpful to clear fractions and move the "parameterized" terms to one side. This gives $$a x^2 + a y^2 - ( a^2 + b^2 + 1 ) x + a = 2 f ( b x - a y )$$ From here, it's pretty clear that the difference of two distinct instances will be a multiple of $0 = bx-ay$, which is therefore the equation of the common radical axis.