C1: Coordinate Geometry in the (x,y) Plane
Equation of a Straight Line
There are many ways of writing an equation to show the relationship between the x- and y-coordinates. The most well-known up to A-level is y = mx + c, but this can sometimes be a bit untidy if there are negatives and fractions. Another common way to write the equation of a straight line is ax + by + c = 0.
We will now look at two different situations when we will be required to find the equation of a straight line. The first is an equation of the line through two points, the second is the equation of the line through a point with a given gradient. I will use the same technique for both of these.
Given 2 Points
Start by finding the gradient of the line joining the points. This is straightforward.
Gradient = (difference in y) / (difference in x).
Remember that you must subtract the y-coordinates in the same order that you subtract the x-coordinates. For example, find the gradient of the line joining (5, 8) to (3, 10)
Once you know the gradient then use the method for finding an equation given a point and the gradient.
Given a Point and the Gradient
The equation of the line has the form y = mx + c. In this form "m" is the gradient, so substitute that in straight away. Now all we need is "c". We find "c" by substituting the x and y coordinates that the line goes through.
For example: find an equation of the straight line through (3, 7) that has gradient 2.
Last example: find an equation in the form ax + by + c = 0, where a, b and c are integers, of the line through (5, 2) and (8, -2)
Note: if you really must work with subscripts (like you might see in a textbook) then please make sure that you understand where the equations come from.
Condition for 2 Lines to be Perpendicular
I won't give reasons why here, but 2 lines are at right angles if their gradients multiply to -1.
Here's an examination question (May 2006 q11, worth 15 marks). Have a bash and then watch the clip to see how well you did.
The line l1 passes through the points P(1, 2) and Q(11, 8).
- Find an equation for l1 in the form y = mx + c, where m and c are constants.
- The line l2 passes through the point R(10, 0) and is perpendicular to l1. The lines l1 and l2 intersect at the point S. Calculate the coordinates of S.
- Show that the length of RS is 3√5
- Hence, or otherwise, find the exact area of the triangle PQR