MPM2D
Ms. Kueh
The Equation of a Circle
A circle is the set of all points in a plane that are the same distance from a fixed point, the centre.
The distance from any point on the circle to the centre is called a radius.
If the centre of the circle is at the origin of the x-y plane and the radius is r units, then
This equation is the equation of a circle with centre (0, 0) and radius r.
Communicate Using the distance from the origin formula, how can you explain why this makes
sense?
Example 1 Write the equation of a circle with centre (0, 0) and a radius of
1
2
Example 2 A circle is defined by the equation 𝑥 2 + 𝑦 2 = 9. Sketch a graph of this circle.
Example 3 A circle has centre (0, 0) and passes through the point (8, −6). Find the equation of the
circle. What are the coordinates of the other point of the diameter that passes through
(8, −6)?
Example 4 A stone is dropped into a pond and sends out a circular ripple whose radius increases by 5
cm/s. Find the equation of the circle 12 s after the stone is dropped.
Example 5 Circle not centred at the origin
a) Find the equation of a circle with radius 4, centred at the point (2, −3). Hint: Use the distance
between two points formula.
b) Find the equation of a circle with radius 9, centred at the point (−5, −10).
Homework Questions:
1. Write the equation of the circle with centre (0, 0) and the given radius, r.
a.
𝑟 = 3
b.
𝑟 = 0.25
c.
1
𝑟 = 23
2. i)
For each circle, state the location or value of the centre, the radius, and the x and y
intercepts.
ii) Graph each circle.
a. 𝑥 2 + 𝑦 2 = 36
b. 𝑥 2 + 𝑦 2 = 49 c. 𝑥 2 + 𝑦 2 = 0.04
3. Find the radius of a circle with centre at (0, 0) that passes through
a.
(−3, 4)
b.
(5, 0)
c.
(0, −3)
d.
(8, −15)
4. For each circle in question 3
i. )
Write the equation
ii. )
Give the coordinates of two other points on the circle
iii.)
Sketch its graph
5. Determine if the point is on, inside, or outside the circle 𝑥 2 + 𝑦 2 = 45. Explain your reasoning.
a. (6, −3)
b.
(−1, 7)
c.
(−3, 5)
6. A rock dropped into a pond sends out a circular ripple whose radius increases steadily at 6
cm/s. A toy boat is floating on the pond 2 m east and 1 m north of the spot where the rock
is dropped. How long does it take for the ripple to reach the boat?
TIPS Practice
1. Points (a, 5) and (9, b) are on the circle 𝑥 2 + 𝑦 2 = 125. What are the values of a and b?
2. Chanelle is creating a design for vinyl flooring. She uses circles and squares to create the
design shown. If the equation of the small circle is 𝑥 2 + 𝑦 2 = 16, what are the dimensions of
the large square?
3. Find the equation of a circle passing through (6, −5) with a radius of 2.
4. Find the equation of a circle whose diameter has endpoints (4, −1) and (−6, 7).
5. Find the distance from the origin to the centre of the circle whose diameter has endpoints (6, 5)
and (2, 1).
6. Circle 1 has a diameter with endpoints (−6, 5) and (−8, 9). Another circle has its centre at one of
the endpoints, and its endpoint at the centre of circle 1. Find all the equations of the possible
circles that fit the description.
Answers:
1. a) 𝑥 2 + 𝑦 2 = 9
2.
b) 𝑥 2 + 𝑦 2 = 0.0625
c) 𝑥 2 + 𝑦 2 =
49
9
a) (0, 0), 6, x-intercepts 6, −6; y-intercepts 6, −6
b) (0, 0), 7, x-intercepts 7, −7; y-intercepts 7, −7
c) (0, 0), 0.2, x-intercepts 0.2, −0.2; y-intercepts 0.2, −0.2
3. a) 5
4.
a)
b)
c)
d)
5.
a) on
b) 5
i) 𝑥 2 + 𝑦 2 = 25
i) 𝑥 2 + 𝑦 2 = 25
i) 𝑥 2 + 𝑦 2 = 9
i) 𝑥 2 + 𝑦 2 = 289
c) 3
ii) (3, 4), (4, 3)
ii) (−5, 0), (0, −5)
ii) (3, 0), (−3, 0)
ii) (0, 17), (0, −17)
b) outside
6. 37.268 s
TIPS practice Answers
1. 𝑎 = 10, 𝑏 = 6.6
2. 11.3 by 11.3
3. (𝑥 − 6)2 + (𝑦 + 5)2 = 4
4. (𝑥 + 1)2 + (𝑦 − 3)2 = 41
5. The circle is 5 units away from the origin.
6. (𝑥 + 8)2 + (𝑦 − 9)2 = 5 and (𝑥 + 6)2 + (𝑦 − 5)2
c) inside
d) 17