Equation of a Circle

•  Edexcel, Module - C2, Chapter - Coordinate Geometry
• AQA, Module - C1, Chapter - Coordinate Geometry
• OCR, Module - C1, Chapter - Coordinate Geometry

In C1, part of the Coordinate Geometry Chapter, we learnt about linear equations and graphing them. In C2, we learn about the equations of circles, from which we can graph them, aswell as solving problems again involving tangents. Remember in GCSE, we learn about Circle Theorems, here are a few we need to know, which will come in use in this chapter :

Circle Terminology !

Circumference - The distance around a circle, given by the formula 2pi * r.

Diameter - The distance from one point on the circumference, to another, passing through the centre of the circle. It is twice the radius (2r)

Radius - A line which is from the centre of the circle, to any point of on the circumference of the circle. It is half the diameter.

Area of a circle is given by the formula - pi * (r^2)

Circle Theorems Recap !

1) The angle in a semicircle is a right angle (An angle formed by drawing lines from the ends of a diameter to its circumference, will form a right angle)

2) The perpendicular from the centre of the chord, bisects the chord.

(A chord is a line that has both endpoints on the circumference on the circle.)

3)The perpendicularity of the radius and tangent.
(A tangent to the circle, forms a right angle, with the circle's radius, at the point of contact with a tagnent).

Showing the 3rd theorem, which shows a line from the centre of the circle till the circumference (Radius), when it meets with a tangent to the circle at a point, it will form a right angle !

Equation of a Circle

The equation of a circle is given in the form : (x-a)² + (y-b)² = r² 

^{a = x -coordinate of centre}

^{b= y-coordinate of centre}

^{r= radius}

<sup>e.g.1 ) What is the equation of a circle when the centre of the circle is (-4,5) and the diameter is 10.

centre of circle (-4,5), so a = -4  and b = 5
radius = half the diameter so 10/2 = 5
so</sup>   (x + 4)² + (y - 5)² = 5<sup>2 


Just leave it in the form 5 squared, rather than writing 25.

e.g.2) What is the centre of the circle, and radius, given this equation of the circle :</sup>

(x)² + (y+8)² = 49


<sup>Centre of Circle -a= 0, and b = 8. So centre of circle is (0, -8 )
Radius is the square root of 49, some people still write + / - 7, even knowing a length can't be negative. so the radius is 7.

We've met our first point, knowing how to derive the equation of a circle, given the centre points and the radius, and vice versa.

Finding the centre of a circle, through Completing the Square


If we are given the equation of the circle to be :</sup>

x² + 6x + y² - 8y - 11 = 0 , and we have to determine the centre of the circle and radius. Here, we can see it's not in the usual form, but still it is an equation of a circle. We just have to complete the square (look under the c1 completing the square post if you've forgotten) :

<sup>complete the square for x terms and y terms seperately:

so</sup>  x² + 6x  becomes = (x+3)² - 9
so y² - 8y becomes = (y - 4)² - 16

we move the -11 to the other side, so we get :

(x+3)² - 9 + (y - 4)² - 16 = 11

(x+3)² + (y - 4)² - 25 = 11

(x+3)² - 9 + (y - 4)² = 36

(x+3)² + (y - 4)² = 6<sup>2

Now, we have to the equation of the cirlce in the desired form :</sup>
(x- a)² + (y - b)² = r<sup>2

The centre of the circle is (-3,4) and the radius is 6.

This is all to the coordinate geometry chapter, in terms of what we need to know. In the next post, I will post some questions and their model solutions that are quite common, which require some knowledge of GCSE theorems!</sup>

 

C2 - Introduction

C2 is the second module, of the Alevel Maths which is core. It is in my opinion the hardest module of AS, for Edexcel, it is one of the longer modules of the whole course.The topics are :

Edexcel

1. Algebra and Functions - Algebraic Division, Factor Theorem, Remainder Theorem.
2. Coordinate Geometry - Equation of Circle, Use of certain circle properties
3. Sequences and Series - Geometric Series {Sum of a g. series, Sum to infinity, Proof of sum formula}, Binomial Expansion for (1+x)^n and (a+b) ^n.
4. Trigonometry - Sine and Cosine Rule, Area of Triangle, Radians, Arc length, area of a sector and segment, Sin,Cos and Tan Graphs, Use of Trig. Identities, Solving simple equations.
5. Exponentials and Logarithms - y=a* and the graph, Laws of Logarithms and Change of base formula.
6. Differentiation - Maxima and Minima problems, Stationery points, Increasing and Decreasing Functions.
7. Integration - Definite Integrals, Area under a curve is a definite integral, Trapezium Rule, Area bounded between a curve and line.

AQA

1. Algebra and Functions - Laws of Indices, Simple transformations on graphs
2. Sequences and Series - Arithmetic series {nth term, Sum of a. series}, Geometric series {Sum of finite g. series, sum to infinity}, Binomial Expansion (1+x)^n.
3. Trigonometry - Sine and Cosine Rule, Area of Triangle, Radians, Arc length, area of a sector and segment, Sin,Cos and Tan Graphs, Use of Trig. Identities, Solving simple equations.
4.  Exponentials and Logarithms - y=a* and the graph, Laws of Logarithms and Change of base formula.
5. Differentiation - Differentiate functions of form x^n. where n is a rational number.
6.  Integration - Integration of functions of form x^n, Trapezium Rule

OCR

1. Algebra - Factor theorem, Remainder theorem, algebraic division, a* graph, laws of logarithms, change of base of logs, and solving log equations.
2. Sequences and Series - Sigma Notation, Arithmetic & Geometric Series, Sum of arithmetic and geometric series, sum to infinity, Binomial Expansion.
3. Trigonometry - Sine and Cosine Rule, Area of Triangle, Radians, Arc length, area of a sector and segment, Sin,Cos and Tan Graphs, Use of Trig. Identities, Solving simple equations.
4.  Integration - Indefinite Integration, Evaluation of definite integrals, Finding area of a region bounded by a curve and line,. Trapezium Rule.

There is alot of new content, and alot which is expanded from the topics learnt in C1. To Achieve a top grade, you should ensure you can apply everything used in C1, these modules are synoptic, and they can ask you a c1 question in c2, or ask a question which relates to c1 content.