The Circle
MPM 2D
Review: 5
Date _________
x
5X5
=25
∴(√25)2=25
"Squaring a Square Root eliminate the square root.
I. The length formula can be used to develop the equation of a circle.
Given the following diagram, let P(x,y) be any point on the circumference of the circle and C(0,0) be the centre of the circle.
Using the length formula:
P(x,y)
r
CP = √(x - 0)2+(y - 0)2
r = √x 2+ y2
Square both side to eliminate the square root
r2 = x2 + y2
C
(0,0)
∴ The equation of a circle with centre C(0, 0) and radius
of r units is:
x2 + y2 = r2
II. Finding the equation of a circle with centre C(0, 0) and r=3: Let P (x, y) represent any point on a circle with radius r = 3 units.
Using the length formula: 3
P(x,y)
CP = √(x - 0)2+(y - 0)2
r =3
3 = √x 2+ y2
Square both side to eliminate the square root
32 = x2 + y2
C
(0,0)
3
∴ The equation of a circle with centre C(0, 0) and radius
of 3 units is:
x2 + y2 = 9
The equation of a circle with centre (0,0) and a radius, 3 is:
x2 + y2 = 9
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Examples:
1. Write an equation for the following circles:
a) Given centre C(0,0) and radius r = 7 units
b) Given centre C(0,0) and radius r = 3.5 units
c) Given centre C(0,0) and radius r = units
Solutions:
a) x2 + y2 = r2
b) x2 + y2 = r2
sub in r = 7
x2 + y2 = 72
2
2
∴ x + y = 49 is the
equation of the circle.
c)
sub in r = 3.5
x2 + y2 = 3.52
2
2
∴ x + y = 12.25 is the
equation of the circle.
x2 + y2 = r2
sub in r = √10
x2 + y2 = (√10)2
2
2
∴ x + y = 10 is the
equation of the circle.
2. Determine the radius of the following circles:
(round to the nearest tenth, if necessary)
a) Given centre C(0,0) and the equation b) Given centre C(0,0) and the equation Solutions:
a)
x2 + y2 = 25
2
∴ r = 25
r = √25
r = 5 units
b)
x2 + y2 = 17
2
∴ r = 17
r = √17
r = 4.1 units
3. Given a circle with centre (0, 0) and passing through the point P(­5, 3):
a) Find the radius
b) Find the equation of the circle
a) x2 + y2 = r2
sub in P(-5, 3) for (x, y)
(-5)2 + (3)2 = r2
25 + 9 = r2
34 = r2
√34 = r
b)
∴The equation of the circle is
x2 + y2 = 34
.
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4. A meteor has crashed into the ocean sending out a circular tidal wave. A ship is located 7 km west and 5 km north of the points of impact. Find the equation of the circle representing the tidal wave when it hits the ship. Homework: 1. Read page 70 Examples 4 and 5
2. page 71 #2, 3, 4, 16
.
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