Equations of circles

Objectives: Students should know the equation of a circle, understand what the equation represents in terms of the circle, and be able to sketch a circle given an equation or find the equation of a circle given the center and a point on that circle.

Lesson:

Begin by having this students perform an investigation of equations of circles in GSP. Have the students draw different circles in GSP and then use the tools in GSP to find the equations of their circles. Then have the students come to a conclusion about what they believe the equations of a circle should be or what properties of circles should factor into its equation. For a copy of this activity click here. For a GSP sketch of this activity click here.

Next, have the students discuss their findings and assumptions as a class

After the students have discussed what they believe should be a part of the equation of a circle, give them the formula:

Given a circle of radius r and center (h, k), the equation of the circle is:

r2 = (x - h)2 + (y - k)2

this is equivalent to saying:

r = sqrt[(x - h)2 + (y - k)2]

where r is the distance from the center of the circle to a point on the circle.

Now provide the students with some examples for finding equations of circles.

What is the equation for a circle of radius 5 and center (3, 7)?

r = 5, h = 3, and k = 7, so

52 = (x - 3)2 + (y - 7)2

25 = (x - 3)2 + (y - 7)2

Given a point (2, 8) on the circle and center (-1, 4) find the equation of the circle.

r is the distance from the center of a circle to any point on the circle, so

r = sqrt[(-1 - 2)2 + (4 - 8)2] = 5

h = -2 and k = 3 therefore

52 = (x + 1)2 + (y - 4)2

25 = (x + 1)2 + (y - 4)2

What do you believe is the equation of the below circle?

the center is (-2, 2) and the point on the circle is (0, 0) therefore the radius must be 2 and h = -2 and k = 2, so we have

22 = (x + 2)2 + (y - 2)2

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

What are the radius and center of the circle whose equation is 64 = (x - 7)2 + (y + 15)2?

the radius is r = sqrt(64) = 8 and the center is (h, k) = (7, -15)