* A circle is the set of all points in a plane that are an equal distance from the centre. The distance from any given point on the circle to the centre is called the radius
* If the centre of the circle is at the origin, (0,0), and the radius is r units, then the equation of the circle is:
or
Example 1
Write the equation of a circle whose center is at the origin (0,0) and has a radius of 2 units.
Solution
The equation of a circle whose centre is at the origin is x² + y² = r²
Since the radius is 2 units long, r² is equal to 4.
Therefore the eqation of this circle is:
x² + y² = 4.
Example 2
A circle has centre (0,0) and passes through point (-8,6). Find the equation of the circle.
Solution
The equation has the form x² + y² = r², because it is centered at the origin.
Here is the graph of the circle.
If we look, more carefully at the circle, we can almost envision a triangle being formed inside the circle:
In order to find the radius, we must use the Pythagorean Theorem.
a² + b² = c²
6² + 8² = c²
36 + 64 = c²
100 = c²
c = 10
Therefore, the radius of the circle is 10 units long.
Therefore, we can say that the equation of the circle is:
x² + y² = 10²
or
x² + y² = 100
Practice Questions
Page 155-156 #1, 3, 4, 5, 9.
More challenging questions: page 157 #11, 12, 14, 15.
