3. The Circle
Also on this page:
a. Circle Formulas
Center at the Origin
The circle with centre (0, 0) and radius r has the equation:
x2 + y2 = r2
This means any point (x, y) on the circle will give the radius squared when substituted into the circle equation.
Center not at the Origin
The circle with centre (h, k) and radius r has the equation:
(x − h)2 + (y − k)2 = r2
These formulas are a direct result of Pythagoras' Formula for the length of the hypotenuse of a right triangle.
Need Graph Paper?
Example 1
Sketch the circle x2 + y2 = 4.
Find the center and radius first.
Example 2
Sketch the circle (x − 2)2 + (y − 3)2 = 16
Find the center and radius first.
Example 3
Sketch the circle (x + 4)2 + (y − 5)2 = 36
b. The General Form of the Circle
An equation which can be written in the following form (with constants D, E, F) represents a circle:
x2 + y2 + Dx + Ey + F = 0
This is called the general form of the circle.
Example 4
Find the centre and radius of the circle
x2 + y2 + 8x + 6y = 0
Sketch the circle.
Exercises
1. Find the equation of the circle with centre `(3/2, -2)` and radius `5/2`.
2. Determine the centre and radius and then sketch the circle:
3x2 + 3y2 − 12x + 4 = 0
3. Find the points of intersection of the circle
x2 + y2 − x − 3y = 0
with the line
y = x − 1.
Conic section: Circle
How can we obtain a circle from slicing a cone?
Each of the lines and curves in this chapter are conic sections, which means the curves are formed when we slice a cone at a certain angle.
If we slice a cone with a plane at right angles to the axis of the cone, the shape formed is a circle.
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