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**Beggarsbelief** Two circles intersect at *p*(2,0) and *q*(-2,8). The distance from the centre of each circle to the common chord [*pq*] is √20

Find the equations of the two circles.

That is the question, I have completed though it took me roughly 40 minutes to complete. I figured out the radius through simple triangle geometry and then created 3 equations in g, f, and c (G being the centre of the circle's x value) (F being the centre of the circle's y value) and c being c.

What resulted was a long drawn out form of substitution which took up one full a4 page to write out and solve, substituting in large equations for f, f^2, c, and c^2 which resulted in getting an equation all in g which eventually led me to a result for g and from there on, substituting back in was easy.

My problem is that it took so long, I am wondering is there another way to solve this question, through trigonometry or did I over complicate it do you think? I am sitting my exams in June and I am expected to answer at least 3 questions of this difficulty along with various other varying difficulty questions, do you think there is an easier way to solve this one than the long, drawn out, painful process I endured, or was what I did correct?