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

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

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

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

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

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

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

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

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

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

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

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