Class: Senior Secondary School 2 (SS2 / SSS2)
Term: 2nd Term
Week: 9
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Coordinate Geometry (Conic Sections)
Previous Lesson: Conic Section: The Circle Meaning and Equation.
Topic: CONIC SECTION: THE CIRCLE
Subject Matter: General equation of a circle, finding centre and radius of a given circle, finding equation of a circle given the end point of the diameter, equation of a circle passing through three points.

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

Define a circle and state its general equation.
Determine the centre and radius of a circle from its general equation.
Derive the equation of a circle given the endpoints of its diameter.
Formulate the equation of a circle passing through three given non-collinear points.

Affective Domain:

Appreciate the application of coordinate geometry in solving real-world problems involving circles.
Show interest in solving problems related to conic sections.

Psychomotor Domain:

Accurately calculate the centre and radius of a circle.
Solve problems to find the equation of a circle using different given conditions.

Collaborate with peers to solve challenging problems on circles.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Schools Education Curriculum
State Unified Scheme of Work
New Further Mathematics Project 2 for Senior Secondary Schools

Instructional Materials

The teacher will teach this lesson with the aid of:

A chart showing the general equation of a circle.
A chart illustrating how to find the equation of a circle passing through 3 points.
Graph papers and rulers.

Rationale for the Lesson

This lesson helps pupils understand the fundamental properties and equations of a circle within the coordinate plane. It enables them to solve various practical problems involving circles, which is important for advanced mathematics and related fields.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of basic coordinate geometry, including distance formula, midpoint formula, and solving simultaneous linear equations.

Lesson Content/Board Summary

The Circle

Definition of a Circle

A circle is the locus of a point that moves in a plane such that its distance from a fixed point (called the centre) is always constant (called the radius).

General Equation of a Circle

The general equation of a circle is given by:

x² + y² + 2gx + 2fy + c = 0

Where the centre of the circle is (-g, -f) and the radius (r) is given by: r = √g² + f² – c

The standard equation of a circle with centre (h, k) and radius r is: (x – h)² + (y – k)² = r²

Example 1: Find the centre and radius of the circle with the equation x² + y² – 6x + 4y – 3 = 0.

Step 1: Compare with the general equation x² + y² + 2gx + 2fy + c = 0.

2g = -6 ⇒ g = -3

2f = 4 ⇒ f = 2

c = -3

Step 2: Find the centre (-g, -f).

Centre = (-(-3), -(2)) = (3, -2)

Step 3: Find the radius r = √g² + f² – c.

r = √(-3)² + (2)² – (-3)

r = √9 + 4 + 3

r = √16

r = 4 units

Therefore, the centre is (3, -2) and the radius is 4 units.

Example 2: Find the centre and radius of the circle (x + 1)² + (y – 5)² = 36.

Step 1: Compare with the standard equation (x – h)² + (y – k)² = r².

h = -1

k = 5

r² = 36 ⇒ r = √36 = 6 units

Step 2: State the centre (h, k) and radius r.

Centre = (-1, 5)

Radius = 6 units

Equation of a Circle Given the Endpoints of its Diameter

If (x₁, y₁) and (x₂, y₂) are the endpoints of a diameter of a circle, the equation of the circle is given by:

(x – x₁)(x – x₂) + (y – y₁)(y – y₂) = 0

Example 3: Find the equation of the circle whose diameter has endpoints A(2, 3) and B(-4, 7).

Step 1: Identify (x₁, y₁) and (x₂, y₂).

Let (x₁, y₁) = (2, 3)

Let (x₂, y₂) = (-4, 7)

Step 2: Substitute into the formula (x – x₁)(x – x₂) + (y – y₁)(y – y₂) = 0.

(x – 2)(x – (-4)) + (y – 3)(y – 7) = 0

(x – 2)(x + 4) + (y – 3)(y – 7) = 0

Step 3: Expand and simplify.

x² + 4x – 2x – 8 + y² – 7y – 3y + 21 = 0

x² + y² + 2x – 10y + 13 = 0

Thus, the equation of the circle is x² + y² + 2x – 10y + 13 = 0.

Equation of a Circle Passing Through Three Points

To find the equation of a circle passing through three non-collinear points, use the general equation x² + y² + 2gx + 2fy + c = 0. Substitute the coordinates of each point into this equation to form three simultaneous linear equations in g, f, and c. Solve these equations to find the values of g, f, and c, then substitute them back into the general equation.

Example 4: Find the equation of the circle passing through the points P(1, 0), Q(0, 1), and R(2, 2).

Step 1: Use the general equation x² + y² + 2gx + 2fy + c = 0.

Step 2: Substitute each point into the equation:

For P(1, 0):

1² + 0² + 2g(1) + 2f(0) + c = 0

1 + 2g + c = 0 (Equation 1)

For Q(0, 1):

0² + 1² + 2g(0) + 2f(1) + c = 0

1 + 2f + c = 0 (Equation 2)

For R(2, 2):

2² + 2² + 2g(2) + 2f(2) + c = 0

4 + 4 + 4g + 4f + c = 0

8 + 4g + 4f + c = 0 (Equation 3)

Step 3: Solve the simultaneous equations.

From (1): c = -1 – 2g

Substitute c into (2):

1 + 2f + (-1 – 2g) = 0

2f – 2g = 0 ⇒ f = g (Equation 4)

Substitute c = -1 – 2g and f = g into (3):

8 + 4g + 4(g) + (-1 – 2g) = 0

8 + 4g + 4g – 1 – 2g = 0

7 + 6g = 0

6g = -7 ⇒ g = -7/6

Since f = g, then f = -7/6

Now find c: c = -1 – 2g = -1 – 2(-7/6) = -1 + 14/6 = -1 + 7/3 = (-3 + 7)/3 = 4/3

So, g = -7/6, f = -7/6, c = 4/3

Step 4: Substitute g, f, and c back into the general equation.

x² + y² + 2(-7/6)x + 2(-7/6)y + 4/3 = 0

x² + y² – (7/3)x – (7/3)y + 4/3 = 0

Multiply by 3 to clear fractions:

3x² + 3y² – 7x – 7y + 4 = 0

This is the equation of the circle.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher reviews the previous lesson on coordinate geometry and asks pupils to recall the distance and midpoint formulas. The teacher then introduces the topic “The Circle” as a conic section.
Pupils’ Activity: Pupils recall previous knowledge and listen attentively to the introduction.
Learning Point: Pupils are reminded of prerequisite knowledge and introduced to the new topic.

Step 2: General Equation and Centre/Radius

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains the definition of a circle and presents the general equation (x² + y² + 2gx + 2fy + c = 0) and the standard equation ((x – h)² + (y – k)² = r²). The teacher explains how to identify the centre and radius from both forms of the equation.
Pupils’ Activity: Pupils listen, take notes, and ask questions for clarification.
Learning Point: Pupils learn the different forms of the circle’s equation and how to find its centre and radius.

Step 3: Worked Examples for General Equation

Time: 8 minutes
Teaching Skill: Problem Solving/Guided Practice
Teacher’s Activity: The teacher works through Example 1 and Example 2 from the board summary, demonstrating step-by-step how to find the centre and radius from given equations.
Pupils’ Activity: Pupils follow the examples, copy the solutions, and participate by suggesting steps.
Learning Point: Pupils practice applying the formulas to find the centre and radius of a circle.

Step 4: Equation from Diameter Endpoints

Time: 8 minutes
Teaching Skill: Explanation/Problem Solving
Teacher’s Activity: The teacher explains how to derive the equation of a circle given the endpoints of its diameter using the formula (x – x₁)(x – x₂) + (y – y₁)(y – y₂) = 0. The teacher then works through Example 3.
Pupils’ Activity: Pupils observe the derivation, copy the formula, and solve the example with the teacher.
Learning Point: Pupils learn to find the equation of a circle when the endpoints of its diameter are given.

Step 5: Equation from Three Points

Time: 8 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains the method for finding the equation of a circle passing through three points, emphasizing the use of simultaneous equations. The teacher then guides pupils through Example 4.
Pupils’ Activity: Pupils pay close attention to the steps involved in setting up and solving the simultaneous equations.
Learning Point: Pupils understand how to determine the equation of a circle given three points.

Step 6: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

State the general equation of a circle.
Find the centre and radius of the circle given by the equation x² + y² + 8x – 2y – 8 = 0.
Determine the equation of the circle whose diameter has endpoints at (1, 1) and (5, 5).
Outline the steps to find the equation of a circle passing through three given points.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 7: Conclusion

Time: 1 minute
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key learning points about the general equation of a circle and methods for finding its equation under different conditions.
Pupils’ Activity: Pupils listen and confirm their understanding.
Learning Point: The lesson is concluded with a recap of the main concepts.

Lesson Keywords

Circle – A round plane figure whose boundary (the circumference) consists of points equidistant from a fixed centre.
Conic Section – A curve obtained as the intersection of a cone with a plane.
Equation of a Circle – An algebraic expression representing all points on the circumference of a circle.
Centre – The fixed point from which all points on the circle are equidistant.
Radius – The constant distance from the centre to any point on the circumference of a circle.
Diameter – A straight line segment that passes through the centre of the circle and whose endpoints lie on the circumference.

Differentiation

For pupils who grasp the concepts quickly, the teacher will provide additional challenging problems involving tangents or normals to a circle. For pupils who require more support, the teacher will provide simplified examples and one-on-one guidance during problem-solving sessions.

Note for teachers using this lesson plan

Ensure pupils have a solid understanding of basic algebra and coordinate geometry before introducing conic sections. Emphasize step-by-step problem-solving and encourage pupils to derive formulas where possible. The use of graphs and visual aids will help in understanding the geometric interpretation of the equations.