# 3. The Circle

General form of a Circle

## 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:

(xh)2 + (yk)2 = r2

These formulas are a direct result of Pythagoras' Formula for the length of the hypotenuse of a right triangle.

### 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 + y2x − 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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