Co-ordinate Geometry of Circle

LCH Paper 2 Q1 The Circle


LCH Paper2 Q1 The Circle

LCHM 2008 II 1ci

2009 p2 q1a

2010 p2 q1 b2

2010 p2 q1 c

Topic Overview

Co-ordinate geometry of the circle (or simply ‘The Circle’) is the second of the co-ordinate topics on our course. It is therefore assumed that students studying The Circle have an adequate working knowledge of the ideas from The Line, the first of the co-ordinate geometry topics. For example, it is taken for granted that students are familiar with the perpendicular distance formula, which is frequently used in The Circle, in connection with questions on tangents. It is also assumed that students are familiar with terms such as tangent and chord, and can recall some of the more elementary geometry theorems from Junior Certificate, e.g., that a radius perpendicular to a chord bisects that chord.

Questions on The Circle generally occur at two levels. At the more straightforward or superficial level, we have to know formulae, learn to recognise the need for them, and substitute values. At the deeper level, information has to be analysed and often a diagram drawn. Then a strategy has to be devised, which may involve four or more separate stages. In such questions, we tend to employ methods rather than relying on formulae.

Topic Structure

The study of Senior Cycle Co-ordinate Geometry of the Circle can be divided into the following sections:

1. Equation of a Circle
We need the radius length to get the equation
If we can get the centre and the radius, we can use this form
The equation of a circle is usually given in this form
We use this method if the centre and the radius cant be found, or are too difficult to find
We can identify g, f, c by comparing with the formula
We find out when we simplify the final equation
Put cos squared plus sin squared = 1
Eliminate t from the equations to get the equation of a circle

2. Two circles
Get the radii from the equations and d from the distance formula
Get the radii from the equations and d from the distance formula

3. Circle and Line
Solve the equations simultaneously
One point of intersection, or perpendicular distance = radius
The proof of this formula should be known
Find the slope of the tangent and then its equation

Let the equation be y=mx+c and solve for m and c

4. Two circles and a line
This is the line that contains the two points of intersection of the circles
This common tangent is perpendicular to the line of centres