Some particular cases of standard equation of a circle

i) Centre is at origin h = 0, and k = 0

x²+y² = a²

(ii) Circle passes through origin
So radius = a² = h²+k²

(x-h)²+(y-k)² = h²+k²

(iii)Circle touches the x axis
C(h,k) centre, a = radius
To satisfy a = k
So equation is
(x-h)²+(y-a)² = a²

(iv)Circle touches the y axis
C(h,k) centre, a = radius
To satisfy a = h
So equation is
(x-a)²+(y-k)² = a²

(v) When the circle touches both axes

then h = k = a
(x-a)²+(y-a)² = a²

(vi) When the circle passes through the origin and centre is on x-axis.
C(h,k) centre, a = radius

As centre is on x axis y coordinate is zero. So k = 0.
As circle is passing through origin a = h
(x-a)²+ y² = a²

(vii) When the circle passes through the origin and centre is on y-axis.
C(h,k) centre, a = radius

As centre is on y axis x coordinate is zero. So h = 0.
As circle is passing through origin a = k
x²+(y-a)² = a²