Circle Equations 2

Circle equations
by

Kristin Vaughn

The standard equation for a circle is:
(x - h)2 + (y - k)2 = r2

Where (h, k) is the center of the circle
and r is the length of the radius.

r
(h, k)

There are several ways that you can be asked a
question that uses this information. The easiest is
when you are given an equation of a circle and
asked to find the length of the radius, the diameter
or the point for the center of the circle.

Find the radius and the center of the following circle.
(x - 9)2 + (y + 3)2 = 36

We compare to the formula:
(x - h)2 + (y - k)2 = r2
(x - 9)2 + (y + 3)2 = 36

As you can see, -h = -9, so h = 9
-k = 3, so k = -3
r2 = 36, so r = 6

The center (h, k) is (9, -3) and the radius r is 6.

Here is a video showing how to find the radius and
center of a circle given the equation. It also shows
how to use this information to graph the circle.
https://www.youtube.com/watch?v=lq7XLnuYSlI

Find the radius and center given the equation
for each of these circles. Turn the page to
check your answers.

1. (x - 5)2 + (y + 1)2 = 100
2. x2 + (y + 2)2 = 25
3. (x + 3)2 + (y + 4)2 = 36
4. x2 + y2 = 64
5. (x - 3)2 + (y - 8)2 = 16

1. (x - 5)2 + (y + 1)2 = 100
Center (5, -1) radius = 10
2. x2 + (y + 2)2 = 25
Center (0, -2) radius = 5
3. (x + 3)2 + (y + 4)2 = 36
Center (-3, -4) radius = 6
4. x2 + y2 = 64
Center (0, 0) radius = 8
5. (x - 3)2 + (y - 8)2 = 16
Center (3, 8) radius = 4

Next we will write an equation for a circle given the
center and the radius.
Given that the center is (-2, 5) and the radius is 3,
write the equation of the circle.
We use the formula (x - h)2 + (y - k)2 = r2, and
plug in the values.

(x - (-2))2 + (y - 5)2 = 32
Then simplify

(x + 2)2 + (y - 5)2 = 9
https://www.youtube.com/watch?v=orsicAAeuT0



Try these three problems, then turn the page to
check your answers.
1. Write the equation for a circle with the center
at (-9, 4) and a radius of 3.

3. What is the equation for a circle that has a center
at (2, -8) and a radius of 1.4?

1. Write the equation for a circle with the center
at (-9, 4) and a radius of 3.

(x + 9)2 + (y - 4)2 = 9

x2 + y2 = 25
3. What is the equation for a circle that has a center
at (2, -8) and a radius of 1.4?

(x - 2)2 + (y + 8)2 = 1.96

Writing the equation of a circle given the center and
a point on the circle.

The point (4, -2) is on a circle whose center is (0, 7).
Write the standard equation of the circle.

First we plug the center into the circle equation:
x2 + (y - 7)2 = r2

Now we plug the point on the circle in for x and y.
42 + (-2 - 7)2 = r2

And solve for r. r = √97 and r2 = 97.
Plug into the equation and you get

x2 + (y - 7)2 = 97

Here is a video to demonstrate finding the
equation given a point on the circle and the
center of the circle.

https://www.youtube.com/watch?v=5dtdupxyQK0

Try these problems and turn to the next
page to check your work.

First plug in the center (x - 4)2 + (y + 2)2 = r2
Then plug in the point for x and y

(5 - 4)2 + (8 + 2)2 = r2 and solve for r.
r = √101 and r2 = 101
and the answer is
(x - 4)2 + (y + 2)2 = 101

(x - 5)2 + (y - 5)2 = 25

Sometimes you will be asked to graph a circle from
an equation.

Example:
Find the radius and the center of the circle, with the
equation (x + 3)2 + (y - 1)2 = 52

Plot the center on a graph and plot 4 points on the
circle. Sketch the circle.

See the video to learn how:

https://youtu.be/MAvd49xAU_Y



Given this equation, find the radius and the center:

(x - 2)2 + (y + 4)2 = 32
Plot the center on a graph. Plot 4 points that
are on the circle.
Check your answer on the next page.

Given this equation, find the radius and the center:

(x - 2)2 + (y + 4)2 = 32
Plot the center on a graph. Plot 4 points that
are on the circle.
The center is (2, -4) and the radius is 3.
You should plot (2, -4) then count 3 up, down,
left and right.

Try these and turn the page to check your answers.
1. Write the equation for a circle with a radius of 5
and the center (-3, 4).
2. Given the circle write the equation.

1. Write the equation for a circle with a radius of 5
and the center (-3, 4).

(x +3)2 + (y - 4)2 = 25
2. Given the circle write the equation.

We see that the center is at
(-2, 3) and the radius is 2, so
(x + 2)2 + (y - 3)2 = 4

Try these and turn the page to check your answers.
= 64.

= 64.
(4, -8) is the center and 8 is the radius

(x - 5)2 + (y + 3)2 = 58
(x - 5)2 + (y - 7)2 = 9

To determine if a point is on a given circle you plug
the point in for x and y to see if it makes the
equation equal.

Given the equation (x - 1)2 + (y + 4)2 = 35 does
(3, 2) fall on the circle?

We plug it in and get (3 - 1)2 + (2 + 4)2 = 35 and we
solve the left to get 40 = 35, which is not correct, so
(3, 2) does not fall on the circle. Also since 40 is
more than 35, we know the point is outside the
circle. (If it was less the point would be inside.)

https://www.youtube.com/watch?v=xradtzJ2Dm4



Try these. Turn the page to check your work.
1. Is the point (0, 4) on the circle

(x + 5)2 + (y - 2)2 = 29?
2. Is the point (1, -3) on the circle

(x - 1)2 + (y + 8)2 = 5?

3. Is the point (-2, 6) on the circle
(x + 4)2 + (y - 5)2 = 5?

4. Is the point (4, 3) on the circle
(x + 3)2 + y2 =58?

1. Is the point (0, 4) on the circle
(x + 5)2 + (y - 2)2 = 29? Yes

2. Is the point (1, -3) on the circle
(x - 1)2 + (y + 8)2 = 5? No, outside

3. Is the point (-2, 6) on the circle
(x + 4)2 + (y - 5)2 = 5? Yes

4. Is the point (4, 3) on the circle
(x + 3)2 + y2 =58? Yes