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).

  1. Find an equation for l1 in the form y = mx + c, where m and c are constants.
  2. 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.
  3. Show that the length of RS is 3√5
  4. Hence, or otherwise, find the exact area of the triangle PQR