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Most of your questions rely on the basic form for a circle:
(x-a)^2 + (y-b)^2 = r^2
with radius r centred at (a,b).
On expanding and tidying up we have
x^2 + y^2 -2ax -2by + a^2 + b^2 - r^2 = 0
So, if we have an equation of the general form
Ax^2 + By^2 + Cxy + Dx + Ey + F = 0
then it could represent a circle only when
A = B <> 0, C = 0
that is, it has the simpler form of
x^2 + y^2 + px + qy + k = 0
Comparing with the standard form of a circle gives
a = -p/2, b=-q/2, and
r^2 = a^2 + b^2 - k = z, say
So, no circle if z <= 0.
You could deduce your own rules to decide how 2 circles interact, according to where their centres C1, C2 are located and their radiuses r1, r2.
For example, if distance between C1 and C2 = r1 + r2, the 2 circles should ...
(x-a)^2 + (y-b)^2 = r^2
with radius r centred at (a,b).
On expanding and tidying up we have
x^2 + y^2 -2ax -2by + a^2 + b^2 - r^2 = 0
So, if we have an equation of the general form
Ax^2 + By^2 + Cxy + Dx + Ey + F = 0
then it could represent a circle only when
A = B <> 0, C = 0
that is, it has the simpler form of
x^2 + y^2 + px + qy + k = 0
Comparing with the standard form of a circle gives
a = -p/2, b=-q/2, and
r^2 = a^2 + b^2 - k = z, say
So, no circle if z <= 0.
You could deduce your own rules to decide how 2 circles interact, according to where their centres C1, C2 are located and their radiuses r1, r2.
For example, if distance between C1 and C2 = r1 + r2, the 2 circles should ...