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)
For a copy of this activity in Word, click here.
Conclusion:
Have the students discuss the solutions to the activity and how they
came to those solutions.
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