Tangents to circles

Figuring out the equations of tangents to circles is often regarded as one of the hardest topics on the GCSE maths syllabus. Combining circle theorems, equations of perpendicular lines and difficult algebraic manipulation, you’ll often find these questions near the back of a paper, and they’re worth quite a bit of marks. However, if you’re confident with your algebra and know the step-by-step technique to approaching these questions, these will be easy marks for you to pick up in your exam. To help you with this, here’s a breakdown of a typical GCSE exam style question on tangents to circles. For example:

The point P (4,3) lies on a circle. Find the equation of the tangent to the circle at point P.

Before we begin any working out, let’s have a look at the given diagram and figure out what we can about the tangent. First of all, we can see that the line is going down, meaning we expect our gradient to be negative. We can also see that the y-intercept is going to be above the highest point of our circle, so we know it should be greater than 5. If we get a final answer where either of these isn’t the case, a mistake has been made….