Definition of circle
The locus of point that moves such that its distance from a fixed point called the center is constant. The constant distance is called the radius, r of the circle.
General Equation (C = A)
From the general equation of conic sections, C = A. Hence, the equation of the circle is
$Ax^2 + Ay^2 + Dx + Ey + F = 0$
$x^2 + y^2 + Dx + Ey + F = 0$
Circle with center at any point (h, k)
Apply Pythagorean Theorem to the figure shown below.
$(x - h)^2 + (y - k)^2 = r^2$
Circle with center at the origin
If the center is at the origin, (h, k) = (0, 0), hence,
$x^2 + y^2 = r^2$