# 3. The Circle

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## a. Circle Formulas

### Center at the Origin

The circle with centre (0, 0) and radius *r* has the equation:

x^{2}+y^{2}=r^{2}

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}=r^{2}

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 *x*^{2} + *y*^{2} = 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**:

x^{2}+y^{2}+Dx + Ey + F= 0

This is called the **general form of the circle**.

### Example 4

Find the centre and radius of the circle

x^{2}+y^{2}+ 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:

3

x^{2}+ 3y^{2}− 12x+ 4 = 0

3. Find the points of intersection of the circle

x^{2}+y^{2}−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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