Skill: Identify the centre and radius from a circle equation

A circle in standard form is written as

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

where:

• $(h,k)$ is the centre
• $r$ is the radius

To identify the centre, look carefully at the signs inside the brackets. The signs are opposite to the coordinates of the centre.

Example

Find the centre and radius of

$$(x+3)^2+(y-4)^2=25.$$

Step 1: Identify the centre.

Compare the equation with

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

For the $x$-part,

$$(x+3)^2=(x-(-3))^2,$$

so $h=-3$.

For the $y$-part,

$$(y-4)^2$$

shows that $k=4$.

Therefore, the centre is

$$\boxed{(-3,4)}.$$

Step 2: Identify the radius.

The number on the right is $r^2$:

$$r^2=25.$$

The radius is the nonnegative square root:

$$r=\sqrt{25}=5.$$

Therefore, the radius is

$$\boxed{5}.$$

So the circle has centre $\boxed{(-3,4)}$ and radius $\boxed{5}$. Remember that a plus sign inside a bracket gives a negative coordinate: $(x+3)^2$ means $x=-3$.