Cogito
Integrated Math 2 · Chapter 6 · Lesson 3
Equations of Circles
A circle written in coordinates.
12 problems · about 21 minutes · G-GPE.A.1
What this lesson teaches
The student writes and interprets the equation of a circle in the coordinate plane.
- A circle is (x − h)² + (y − k)² = r², with centre (h, k).
- The right side is r squared, so take the root for the radius.
- Completing the square recovers the centre from an expanded equation.
Warm Up
Straightforward practice. Get the method working first.
5 problems(x − 3)² + (y − 2)² = 16. What is the radius?
Answer 4
Why 4.
Where does the circle equation come from?
Answer The distance formula, applied to a fixed distance from a centre.
Why It is the distance formula, squared.
x² + y² = 36. What is the radius?
Answer 6
Why √36.
(x − 4)² + y² = 25. Radius?
Answer 5
Why √25.
Same circle. Centre x-coordinate?
Answer 4
Why The sign reverses.
Build It Up
The same ideas with more to keep track of.
3 problems(x + 5)² + (y − 1)² = 9. Centre x-coordinate?
Answer -5
Why x + 5 is x − (−5).
To complete the square on x² − 6x, what is added?
Answer 9
Why (−3)².
x² + y² = 0. How many points does the graph contain?
Answer 1
Why Only the origin.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the steps for finding a centre from general form in order.
Answer 1. Group the x terms and the y terms. 2. Move the constant to the right. 3. Complete the square on each group, adding to both sides. 4. Read the centre and take the root of the right side.
Why The grouping happens first.
(x − 1)² + (y + 3)² = 64. Radius?
Answer 8
Why √64.
The Radius: (x − 2)² + (y − 5)² = 49. What is the radius?
Answer 7
Why 7.
The Centre: Same circle. What is the x-coordinate of the centre?
Answer 2
Why 2.