Circle Problems- Sketch it out!

Many past papers and practice papers incorporate Circle problems but very few students really push themselves to visualise the problem.

I will show 4 examples from a set of practice papers that highlight this issue. The question is from a CGP practice paper and are from 4 different tutees who live in different parts of the country (hence complete independence).

Part (a) of the question involved finding the centre (X) and radius r. The centre is (4,-2) and radius is 7 which most students figured out very well.

Part (b) however across the board was an issue, as you can see below:

Student 1- tried to do a half hearted sketch in top corner

Student 2- slightly better sketch but not accurate

Student 3- drew a decent sketch but was not able to see the position of P in context

Student 4- the final answer is correct BUT the sketch and the maths shown do not link!

Overall it is very interesting to see the approach of my 4 tutees for this question.

Solution- Sketch the circle and read the question!

First of all I would advise students to draw a VERY ROUGH sketch of the circle, it does not matter exactly where the circle sits relative to the x,y axis for now as we want to visualise the problem with the tangent involved. There are however situations where you need to be more precise and you should take your time in those problems.

Once this has been drawn I had to read the question, particularly that the tangent FROM the point P(-16,13). This means the tangent starts from P and touches the circle at a point Y( which we do not know).

Once you have drawn the tangent you can see a clear right angled triangle problem as the tangent is at a right angle to the radius. You know the radius from part (a) and you can use the length between 2 co-ordinates formula to find the length of PX. Once done you have found the hypotenuse and can use this to get the length of PY (shorter side) using simple Pythagoras theorem.

It also helps to label what lengths you are finding, I like to use capital L and subscripts to denote the length.

Comparison with not sketching

You can clearly see by sketching the circle it helps to:

• Visualise the problem

• See the bigger picture (i.e. that it is a right angled triangle problem)

• Improve your graph sketching ability with different graphs (circle and tangent in this case).

• Supplement your mathematical working by providing a firm base to 'attack' the problem.

Conclusion:

If circles is a topic you struggle with I would highly suggest you find questions in your textbook where you are not given a diagram and have to visualise the problem in order to find the answer.

There are a few example of this online, for example this link.

I wish you the best and remember: If in doubt sketch it out!