Skill: Determine the equation of a circle from its centre and radius

A circle with centre $(h,k)$ and radius $r$ has equation

$$(x-h)^2+(y-k)^2=r^2.$$

The signs are important: you subtract the coordinates of the centre. If a coordinate is negative, subtracting it produces a plus sign.

Example: Determine the equation of the circle with centre $(-2,3)$ and radius $5$.

1. Identify the values:

$$h=-2,\qquad k=3,\qquad r=5$$

2. Substitute them into the standard form:

$$(x-(-2))^2+(y-3)^2=5^2$$

3. Simplify:

$$\boxed{(x+2)^2+(y-3)^2=25}$$

Therefore, the equation of the circle is

$$\boxed{(x+2)^2+(y-3)^2=25}.$$

Notice that the centre is $(-2,3)$, even though the equation contains $(x+2)$. The plus sign appears because $x-(-2)=x+2$.