P2 Chapter 2 :: Circles
Chapter Overview
From this chapter onwards, the majority of the theory you will learn is new since GCSE.
1:: Equation of a circle
3:: Chords, tangents and perpendicular bisectors.
4:: Circumscribing Triangles
2:: Intersections of lines + circles
Midpoints and Perpendicular Bisectors
Later in the chapter you will need to find the perpendicular bisector of a chord of a circle.
chord
perpendicular bisector
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Test Your Understanding
a
b
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Fro Note: Do not try to memorise this!
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Exercise 2A/2B
Pearson Pure Mathematics Text book
Page 27,28,29
Equation of a circle
(Hint: draw a right-angled triangle inside your circle, with one vertex at the origin and another at the circumference)
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Equation of a circle
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Quickfire Questions
Centre
Radius
Equation
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Finding the equation using points
Hint: What two things do we need to use the circle formula?
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Test Your Understanding
Edexcel C2 Jan 2005 Q2
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Completing the square
Complete the square!
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Textbook Note: There’s a truly awful method, initially presented in the textbook, that allows you to find the centre/radius without completing the square. Don’t even contemplate using it.
Further Example
Edexcel C2 June 2012 Q3a,b
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Exercise 2C
Pearson Pure Mathematics Year
Page 31
Extension:
1
1
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1
2
3
Intersections of Lines and Circles
Recall that to consider the intersection of two lines, we attempt to solve them simultaneously by substitution, potentially using the discriminant to show that there are no solutions (and hence no points of intersection).
2 intersections (such a line is known as a secant of the circle)
1 intersections (such a line is known as a tangent of the circle)
0 intersections
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Test Your Understanding
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Exercise 2D
Pearson Pure Mathematics
Page 34
Tangents, Chords, Perpendicular Bisectors
radius
tangent
chord
perpendicular bisector
The tangent is perpendicular to the radius (at the point of intersection).
The perpendicular bisector of any chord passes through the centre of the circle.
There are two circle theorems that are of particular relevance to problems in this chapter, the latter you might be less familiar with:
Why this will help:
If we knew the centre of the circle and the point of intersection, we can easily find the gradient of the radius, and thus the gradient and hence equation of the tangent.
Why this will help:
The first thing we did in this chapter is find the equation of the perpendicular bisector. If we had two chords, and hence found two bisectors, we could find the point of intersection, which would be the centre of the circle.
Examples
Note that the GCSE 2015+ syllabus had questions like this, except with circles centred at the origin only.
Fro Tip: Use ‘subscripting’ of variables to make clear to the examiner (and yourself!) what you’re calculating.
This time we have the gradient, but don’t have the points where the tangent(s) intersect the radius.
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Determining the Circle Centre
a
b
c
Fro Exam Tip: If you’re not asked for the equation in a particular form, just leave it as it is.
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Test Your Understanding
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Exercise 2E
Pearson Pure Mathematics Text Book
Pages 38-40
Extension:
Mark scheme on next slide.
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2
3
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(This is not a tangent/chord question but is worthwhile regardless!)
Mark Scheme for Extension Question 2
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Mark Scheme for Extension Question 3
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Triangles in Circles
We’d say:
Triangles in Circles
Given three points/a triangle we can find the centre of the circumcircle by:
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Example
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b) Hence find the equation of the circle.
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Example
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Exercise 2F
Pearson Pure Mathematics
Pages 43-44
Extension:
1
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