3.2.2
Finding the Centre & Radius
How do you find the Centre and Radius of a Circle?
How do you find the Centre and Radius of a Circle?
To find the centre and radius of a circle from its expanded equation, we need to complete the square.
What is the expanded equation of a circle?
What is the expanded equation of a circle?
- If we expand the general equation of a circle with centre and radius
we get:
What is the expanded equation of a circle?
What is the expanded equation of a circle?
- If we make the substitutions:
- Then the expanded form of the equation of a circle is given by:
How do you find the centre and radius of a circle?
How do you find the centre and radius of a circle?
- If the equation of a circle is in the expanded form, we can complete the square and find the circle in the general form:
- .
- The centre of the circle is at the point and the radius is of length .
Example
Example
- What is the centre and radius of the circle with equation ?
Group together and
terms
Group together and terms
- To begin, we group together the terms containing and the terms containing
:
Complete the square for terms containing
Complete the square for terms containing
- We then complete the square for the terms alone:
Complete the square for terms containing
Complete the square for terms containing
- We then complete the square for the terms alone:
Substitute in
Substitute in
- We then substitute these back into the equation:
Simplify
Simplify
- Simplifying the constants gives:
- Then adding 9 (which is 32) to each side, we get the equation of the circle in its general form:
Inspect
Inspect
- Comparing this to the general form of the equation, we can see that the centre of the circle is at the point (2,3) and the radius is equal to 3.
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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