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Straight Lines

You will be expected to be able to find the equation of a straight line given a variety of scenarios. This could be as simple as finding the equation of a line given two points, or it could be finding the equation of a tangent or a normal or even using perpendicularity to find the equation of a straight line.

The equation of a line given two points

Recall that to find the equation of a straight line between two points, you must first find the gradient of the line that connects the two points. This gradient is usually calculated using rise over run. One of the points can then be used to calculate the y-intercept of the line.

Example: find the equation of the line between the points (2,4) and (4,10). – Imagine the line that connects these two points – it rises by 6 as it runs along 2 and so rise over run is 6/2 which equals 3 and so the gradient of the line is 3. This tells us that the line is of the form y=3x+c. To find c, plug one of the given points into the equation so far: 10=3\times 2 +c and so c must be 4 giving the final equation of the line as y=3x+4.

Tangents and Normals

In order to find the equation of a tangent or normal to a curve, one must be able to find the gradient of a curve and this required knowledge or differentiation. See Tangent & Normals.

Perpendicularity

You may be required to find the equation of a straight line when it might not be possible to calculate the gradient of a line directly. The gradient of a line that is perpendicular to it may be given in which case if the gradient of this line is m then the gradient of the required line is -\frac{1}{m}.
It might be that you are given two points on a line that is perpendicular and you will have to calculate m first or you might be given other characteristics, such as a tangent to a circle which you know to be perpendicular to the radius.