1. Coordinate Geometry – The Circle
This chapter will focus on solving problems involving circles.
We will be:
Finding the distances between points
Finding the midpoint of a line
Finding the equation of a circle
Finding points of intersection of a circle and a line

2. Coordinate Geometry – The Circle
CONTENTS
Required Pre-Knowledge
Equation of a Line
Circle Terminology
Completing the Square
Distance Between Two Points
New Material
Finding the Midpoint of a Line
Example 1
Example 2
Example 3
Example 4
Assignment 2

3. Coordinate Geometry – The Circle
REQUIRED PRE-KNOWLEDGE
Equation and Gradient of a Line
For the equation of a line we need either a point and the gradient or two points.
Gradient of line = (y2 – y1)/(x2 – x1)
Gradients of perpendicular lines multiplied together equals -1 3

4. Coordinate Geometry – The Circle
REQUIRED PRE-KNOWLEDGE
Terminology of Circles
Circle: A circle is a shape drawn by keeping a pencil the same distance from a fixed point on a piece of paper.
Circumference: The circumference is the perimeter of the circle.
Radius: The distance from the centre of the circle to any point on the circumference.
Diameter: The distance across the circle, passing through the centre. The diameter is twice the length of the radius. 4

5. Coordinate Geometry – The Circle
REQUIRED PRE-KNOWLEDGE
Complete the Square
Given a quadratic such as x2 + bx you should be able to complete the square using the formula 5

6. Coordinate Geometry – The Circle
REQUIRED PRE-KNOWLEDGE
Distance Between Two Points
Given two points (x1, y1) and (x2, y2) we can find the distance between the two points using the formula 6

7. Coordinate Geometry – The Circle
Finding the Midpoint of a Line
We can find the midpoint of a line segment joining two points, (x1, y1) and (x2, y2) using the formula: 7

8. Coordinate Geometry – The Circle
Example 1
Find the midpoint of the line joining A(3, 4) and B(-1, 6).
Solution:
We substitute into the formula for the midpoint of a line -
- to get: 8

9. Coordinate Geometry – The Circle
Example 2
Find the centre of the circle given A(0, 3) and B(4, 3) are endpoints of the diameter.
Solution:
The midpoint of a circle’s diameter is the centre of the circle.
Therefore if we find the coordinates of the midpoint of the line joining A and B we will have the centre of the circle.
Substitute into the formula for the midpoint of a line to get: 9

10. Coordinate Geometry – The Circle
Example 3
The line AB is a diameter of a circle, where A and B are (-3, -4) and (6, 10) respectively. Show that the centre of the circle lies on the line y = 2x.
Solution:
We first need to find the coordinates of the centre of the circle using the midpoint formula. This gives:
We then substitute our x coordinate into the equation y = 2x and if the answer equals the y coordinate then the point lies on the line. This gives:
The y value equals the y coordinate, therefore the point does lie on the line y = 2x. 10

11. Coordinate Geometry – The Circle
Example 4
The line AB is the diameter of a circle with centre O. Given A(1, 3) and O(3, 5) find the coordinates of B.
Solution:
We know one endpoint of the diameter and the midpoint of the diameter (the centre O). We can substitute these values into the midpoint formula with (x, y) as the coordinates of B.
This will give two equations to solve. Using midpoint formula gives:
Coordinates of B are (5, 7) 11

12. Coordinate Geometry – The Circle
Week 1 – Assignment
The assignment this week is a Moodle Assignment. Follow the link in the Course Area.
Deadline for submission of assignments is
5:00pm, Monday 8 February 2010
No late submissions will be accepted. 12