A circle centred at the origin with radius $r$ has equation: $$x^2 + y^2 = r^2$$
A circle centred at point $(a, b)$ with radius $r$ has equation: $$(x - a)^2 + (y - b)^2 = r^2$$
Example: A circle with centre $(3, -2)$ and radius 5: $$(x - 3)^2 + (y + 2)^2 = 25$$
Finding the radius and centre from an equation: compare with the standard form. $$x^2 + y^2 = 49 \implies \text{centre } (0,0), \text{ radius } 7$$
Tangent to a circle: a tangent at point $(x_1, y_1)$ on circle $x^2 + y^2 = r^2$ is: $$x\cdot x_1 + y\cdot y_1 = r^2$$
The tangent is perpendicular to the radius at the point of contact.
Common error: writing $(x + a)^2 + (y + b)^2 = r^2$ for centre $(a, b)$ — remember the signs flip.