Lets start with how to write an equation of a circle in general. There are two pieces of information we need all the time to write and equation of a circle. Those two things are the center and the radius. Once we have the center and radius, it is a matter of "plugging" the numbers in the appropriate places in the formula. Recall that the formula for an equation of a circle is (x - h)² + (y - k)² = r² where (h, k) is the center and r is the radius.

So if we are given the center is (3,0) and the radius is 4, can you determine the equation? Try it and check below.

The equation should be:

(x - 3)² + (y-0)² = 4²

So this simplifies to a final answer of:

(x-3)² + y² = 16

Sometimes the information given isn't as straightforward as here is the center and here is the radius. But enough information is given so that we can use a sketch or our "brain" to find the center and the radius. So we MUST find the center and radius, then write our equation.

For example, lets say the center is given to be (3,1) but instead of giving us a radius, the information that is given is that the circle goes through the point (5,4). By sketching this information, either on paper or in our head, we should realize that the distance from the center to any point on the circle will give us the radius, so we must find the length of the radius by using the distance formula. Do you remember what that is? Try and if you don't remember, look down below.

d = √((x₂ -x₁)² + (y₂ - y₁)²)

Therefore the distance = √((5 - 3)² + (4 - 1)²) which gives us √(2² + 3²), giving us a final radius of √(4 + 9)= √13.

Now see if you can use this information to write your equation of a circle. Check it below.

(x - 3)² + (y-1)² = √13²

Which is a final answer of: (x-3)² + (y-1)² = 13

Lets try one more. Suppose we are given the radius of a particular circle is 4 and the circle has its center in the first quadrant and is tangent to the y-axis at (0,2) If we make a sketch of this information, we can see by plotting the point (0,2) and sketching a circle just touching the y axis at that point (definition of a tangent from geometry) that the center must be at (4,2).

Using all of this information, can you write the equation of the circle now? Remember all you use is the center and radius. Check below.

The final equation should be :

(x - 4)² + (y - 2)² = 16