MTH5P1: Co-ordinate geometry

 Cordinate geometry

Co-ordinates

Co-ordinates are a means of describing a position relative to some fixed point, or origin. In two dimensions you need two pieces of information; in three dimensions, you need three pieces of information.
In the Cartesian system (named after René Descartes), position is given in perpendicular directions: x, y in two dimensions; x, y, z in three dimensions (see figure 2.1). This chapter concentrates exclusively on two dimensions.

Plotting, sketching and drawing

In two dimensions, the co-ordinates of points are often marked on paper and joined up to form lines or curves. A number of words are used to describe this process.

Plot (a line or curve) means mark the points and join them up as accurately as you can. You would expect to do this on graph paper and be prepared to read information from the graph.

Sketch means mark points in approximately the right positions and join them up in the right general shape.

You would not expect to use graph paper for a sketch and would not read precise information from one.

You would however mark on the co-ordinates of important points, like intersections with the x and y axes and points at which the curve changes direction.

Draw means that you are to use a level of accuracy appropriate to the circumstances, and this could be anything between a rough sketch and a very accurately plotted graph.

The gradient of a line

In everyday English, the word line is used to mean a straight line or a curve. In mathematics, it is usually understood to mean a straight line.

If you know the coordinates of any two points on a line, then you can draw the line.
The slope of a line is measured by its gradient.

It is often denoted by the letter m. In figure 2.2, A and B are two points on the line.

The gradient of the line AB is given by the increase in the y coordinate from A to B divided by the increase in the x coordinate from A to B.

When the same scale is used on both axes, m = tan θ (see figure 2.2). Figure 2.3 shows four lines.

Looking at each one from left to right: line A goes uphill and its gradient is positive; line B goes downhill and its gradient is negative.

Line C is horizontal and its gradient is 0; the vertical line D has an infinite gradient.

Parallel and perpendicular lines

Parallel lines have the same gradient

The product of the gradient of the perpendicular lines is equal to -1

The distance between two points

When the coordinates of two points are known, the distance between them can be calculated using Pythagoras’ theorem

This method can be generalised to find the distance between any two points,

The mid-point of a line joining two points

The formula for the coordinates of the mid points between two points is given by

A and B are the points (2, 5) and (6, 3) respectively (see figure 2.9). Find:

(i) the gradient of AB
(ii) the length of AB
(iii) the mid-point of AB
(iv) the gradient of a line perpendicular to AB.

SOLUTION

Using two different methods, show that the lines joining P(2, 7), Q(3, 2) and R(0, 5) form a right-angled triangle (see figure 2.10).

SOLUTION

Method 2
Pythagoras’ theorem states that for a right-angled triangle whose hypotenuse has length a and whose other sides have length

The equation of a straight line

The word straight means going in a constant direction, that is with fixed gradient.
This fact allows you to find the equation of a straight line from first principles.

Find the equation of the straight line with gradient 2 through the point (0, −5).

Lines parallel to the axes
Lines parallel to the x axis have the form y = constant, those parallel to the y axis the form x = constant. Such lines are easily recognised and drawn.

Equations of the form y = mx + c
The line y = mx + c crosses the y axis at the point (0, c) and has gradient m. If c = 0, it goes through the origin. In either case you know one point and can complete the line either by finding one more point, for example by substituting x = 1, or by following the gradient (e.g. 1 along and 2 up for gradient 2).

Equations of the form px + qy + r = 0
In the case of a line given in this form, like 2x + 3y − 6 = 0, you can either rearrange it in the form y = mx + c

SOLUTION
The line x = 5 is parallel to the y axis and passes through (5, 0).
The line y = 0 is the x axis.
The line y = x has gradient 1 and goes through the origin.

The triangle obtained is an isosceles right-angled triangle, since OA = AB = 5
units, and ∠OAB = 90°.
EXAMPLE 2.5 Draw y = x − 1 and 3x + 4y = 24 on the same axes.
SOLUTION
The line y = x − 1 has gradient 1 and passes through the point (0, −1).
Substituting y = 0 gives x = 1, so the line also passes through (1, 0).
Find two points on the line 3x + 4y = 24.
Substituting x = 0 gives 4y = 24 so y = 6.
Substituting y = 0 gives 3x = 24 so x = 8.

The line passes through (0, 6) and (8, 0).

Finding the equation of a line

The simplest way to find the equation of a straight line depends on what information you have been given.

Given the gradient, m, and the co-ordinates (x, y) of one point on the line

Find the equation of the line with gradient 3 which passes through the point (2, −4).
SOLUTION

Find the equation of the line joining (2, 4) to (5, 3).

SOLUTION

The intersection of two lines

The intersection of any two curves (or lines) can be found by solving their equations simultaneously. In the case of two distinct lines, there are two possibilities:

they are parallel
they intersect at a single point.

Sketch the lines x + 2y = 1 and 2x + 3y = 4 on the same axes, and find the coordinates of the point where they intersect.

SOLUTION

The coordinates of the point of intersection are (5, −2).

Find the co-ordinates of the vertices of the triangle whose sides have the
equations x + y = 4, 2x − y = 8 and x + 2y = −1.
SOLUTION
A sketch will be helpful, so first find where each line crosses the axes.
1.  x + y = 4 crosses the axes at (0, 4) and (4, 0).
2.  2x − y = 8 crosses the axes at (0, −8) and (4, 0).

Since two lines pass through the point (4, 0) this is clearly one of the vertices. It has been labelled A on figure 2.24.
Point B is found by solving 2 and 3 simultaneously:

The line l has equation 2x − y = 4 and the line m has equation y = 2x − 3.
What can you say about the intersection of these two lines?

René Descartes was born near Tours in France in 1596. At the age of eight he was sent to a Jesuit boarding school where, because of his frail health, he was allowed to stay in bed until late in the morning. This habit stayed with him for the rest of his life and he claimed that he was at his most productive before getting up.
After leaving school he studied mathematics in Paris before becoming in turn a soldier, traveller and optical instrument maker. Eventually he settled in Holland where he devoted his time to mathematics, science and philosophy, and wrote a number of books on these subjects.
In an appendix, entitled La Géométrie, to one of his books, Descartes made the contribution to co-ordinate geometry for which he is particularly remembered.
In 1649 he left Holland for Sweden at the invitation of Queen Christina but died there, of a lung infection, the following year.

The intersection of a line and a curve

When a line and a curve are in the same plane, there are three possible situations.