Class: Senior Secondary School 2 (SS2 / SSS2)
Term: 2nd Term
Week: 10
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Coordinate Geometry / Conic Sections
Previous Lesson: Conic Section: General Equation of a Circle and Applications.
Topic: CONIC SECTION: THE CIRCLE
Subject Matter: Equation of tangent to a circle, length of tangent to a circle

Specific Objectives

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

Cognitive Domain:

Define a tangent to a circle.
Recall the standard equations of a circle.
Derive or state the formula for the equation of a tangent to a circle at a given point.
Calculate the length of a tangent from an external point to a circle.

Affective Domain:

Appreciate the application of tangents in various mathematical and real-world contexts.
Show interest in solving problems related to tangents and circles.

Psychomotor Domain:

Accurately find the equation of a tangent to a circle using appropriate formulas or methods.
Accurately calculate the length of a tangent from an external point to a circle.

Collaborate with peers to solve problems involving tangents to a circle.
Share different approaches to solving tangent-related problems.

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:

Chart showing the tangent of a circle and the length of a tangent.
Whiteboard and markers.
Geometric instruments (compass, ruler).

Rationale for the Lesson

This lesson enables pupils to understand how to find the equation of a line that touches a circle at exactly one point and how to calculate the distance from an external point to the circle along such a line. This knowledge is important for further studies in coordinate geometry and calculus.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of the standard equations of a circle, gradients of lines, and basic differentiation.

Lesson Content/Board Summary

CONIC SECTION: THE CIRCLE

Equation of Tangent to a Circle

A tangent to a circle is a straight line that touches the circle at exactly one point. This point is called the point of tangency. The radius drawn to the point of tangency is perpendicular to the tangent.

1. For a circle with center at the origin (0, 0) and equation x² + y² = r²:
The equation of the tangent at a point (x₁, y₁) on the circle is given by:

x x₁ + y y₁ = r²

Worked Example 1: Find the equation of the tangent to the circle x² + y² = 25 at the point (3, 4).

Step 1: Identify the given values.
Circle equation: x² + y² = 25 (so r² = 25)
Point of tangency (x₁, y₁) = (3, 4)

Step 2: Apply the formula x x₁ + y y₁ = r².
x(3) + y(4) = 25
3x + 4y = 25

The equation of the tangent is 3x + 4y = 25.

2. For a circle with center (a, b) and equation (x-a)² + (y-b)² = r²:
The equation of the tangent at a point (x₁, y₁) on the circle is given by:

(x – a)(x₁ – a) + (y – b)(y₁ – b) = r²

Alternatively, using differentiation:

Step 1: Differentiate the circle’s equation implicitly to find dy/dx (gradient of the tangent).
Step 2: Substitute the coordinates (x₁, y₁) into dy/dx to get the gradient ‘m’ of the tangent.
Step 3: Use the point-slope form of a straight line: y – y₁ = m(x – x₁).

Worked Example 2: Find the equation of the tangent to the circle (x-1)² + (y+2)² = 10 at the point (4, -1).

Method 1: Using the direct formula
Step 1: Identify the given values.
Circle equation: (x-1)² + (y+2)² = 10 (so a=1, b=-2, r²=10)
Point of tangency (x₁, y₁) = (4, -1)

Step 2: Apply the formula (x – a)(x₁ – a) + (y – b)(y₁ – b) = r².
(x – 1)(4 – 1) + (y – (-2))(-1 – (-2)) = 10
(x – 1)(3) + (y + 2)(1) = 10
3x – 3 + y + 2 = 10
3x + y – 1 = 10
3x + y = 11

The equation of the tangent is 3x + y = 11.

Length of Tangent from an External Point to a Circle

The length of the tangent from an external point P(x₁, y₁) to a circle with equation x² + y² + 2gx + 2fy + c = 0 is given by:

L = √ (x₁² + y₁² + 2gx₁ + 2fy₁ + c)

If the circle equation is in the form (x-a)² + (y-b)² = r², first expand it into the general form or substitute the point directly into the left side and subtract r²:

L = √ ((x₁-a)² + (y₁-b)² – r²)

Worked Example 3: Find the length of the tangent from the point (5, 1) to the circle x² + y² – 4x + 6y – 12 = 0.

Step 1: Identify the given values.
External point (x₁, y₁) = (5, 1)
Circle equation: x² + y² – 4x + 6y – 12 = 0

Step 2: Substitute (x₁, y₁) into the formula L = √ (x₁² + y₁² + 2gx₁ + 2fy₁ + c).
L = √ (5² + 1² – 4(5) + 6(1) – 12)
L = √ (25 + 1 – 20 + 6 – 12)
L = √ (26 – 20 + 6 – 12)
L = √ (6 + 6 – 12)
L = √ (12 – 12)
L = √0
L = 0

NOTE: A length of 0 means the point (5,1) is on the circle, not external.

Worked Example 4: Find the length of the tangent from the point (7, -1) to the circle (x-2)² + (y+3)² = 9.

Step 1: Identify the given values.
External point (x₁, y₁) = (7, -1)
Circle equation: (x-2)² + (y+3)² = 9 (so a=2, b=-3, r²=9)

Step 2: Apply the formula L = √ ((x₁-a)² + (y₁-b)² – r²).
L = √ ((7 – 2)² + (-1 – (-3))² – 9)
L = √ ((5)² + (-1 + 3)² – 9)
L = √ (25 + (2)² – 9)
L = √ (25 + 4 – 9)
L = √ (29 – 9)
L = √20

Step 3: Simplify the surd.
L = √ (4 × 5)
L = 2√5 units

The length of the tangent is 2√5 units.

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 concept of a circle and its equations. The teacher then introduces the concept of a tangent line to a circle, relating it to real-life examples like a wheel touching the ground.
Pupils’ Activity: Pupils recall circle equations and listen attentively to the introduction of tangents.
Learning Point: Pupils recall previous knowledge and are introduced to the lesson’s topic.

Step 2: Equation of Tangent (Method 1: Direct Formula)

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher guides pupils to state the formula for the equation of a tangent to a circle x² + y² = r² at a point (x₁, y₁). The teacher demonstrates a worked example from the board summary.
Pupils’ Activity: Pupils write down the formula and follow the steps of the worked example, asking questions for clarification.
Learning Point: Pupils learn how to find the equation of a tangent for a circle centered at the origin.

Step 3: Equation of Tangent (Method 2: General Circle)

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher guides pupils through the formula for the equation of a tangent to a circle (x-a)² + (y-b)² = r² at a point (x₁, y₁). The teacher demonstrates a worked example from the board summary, explaining the alternative method using differentiation.
Pupils’ Activity: Pupils copy the formula and the worked example, understanding the application for a general circle.
Learning Point: Pupils learn to find the equation of a tangent for a circle with any center.

Step 4: Practice Problems (Tangent Equation)

Time: 5 minutes
Teaching Skill: Problem Solving
Teacher’s Activity: The teacher provides a few practice problems for pupils to solve on finding the equation of a tangent. The teacher monitors and offers assistance.
Pupils’ Activity: Pupils attempt to solve the practice problems individually or in small groups.
Learning Point: Pupils apply the learned formulas and methods to solve problems.

Step 5: Length of Tangent from an External Point

Time: 5 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher introduces the concept of the length of a tangent from an external point to a circle. The teacher states the formula and demonstrates a worked example from the board summary.
Pupils’ Activity: Pupils listen, copy the formula, and observe the steps for calculating the length of a tangent.
Learning Point: Pupils understand how to calculate the length of a tangent from an external point.

Step 6: Practice Problems (Length of Tangent)

Time: 5 minutes
Teaching Skill: Problem Solving
Teacher’s Activity: The teacher gives pupils practice questions on finding the length of a tangent from an external point. The teacher provides guidance as needed.
Pupils’ Activity: Pupils work on the practice problems to reinforce their understanding.
Learning Point: Pupils practice calculating the length of a tangent from an external point.

Step 7: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

Define a tangent to a circle.
State the formula for the equation of the tangent to the circle x² + y² = r² at the point (x₁, y₁).
Find the equation of the tangent to the circle x² + y² = 13 at the point (2, 3).
Calculate the length of the tangent from the point (6, 2) to the circle x² + y² – 2x – 4y – 4 = 0.

Pupils’ Activity: Pupils answer orally and in writing.
Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 5 minutes
Teaching Skill: Summarizing
Teacher’s Activity: The teacher summarizes the key learning points on finding the equation of a tangent and the length of a tangent. The teacher assigns homework involving various problems on the topic.
Pupils’ Activity: Pupils take note of the summary and copy the assigned homework.
Learning Point: Pupils consolidate their understanding and prepare for independent practice.

Lesson Keywords

Tangent – A line that touches a curve at exactly one point.
Circle – A round plane figure whose boundary (the circumference) consists of points equidistant from a fixed central point.
Equation – A statement that the values of two mathematical expressions are equal.
Length of Tangent – The distance from an external point to the point of tangency on a circle.
Point of Tangency – The single point where a tangent line touches a circle.

Differentiation

For pupils who grasp the concepts quickly, the teacher will provide more complex problems involving finding the tangent when the point is not given or when the tangent is parallel/perpendicular to another line. For pupils who require more support, the teacher will provide simpler problems and offer one-on-one guidance, reinforcing the basic formulas and step-by-step procedures.

Note for teachers using this lesson plan

Ensure pupils have a strong foundation in coordinate geometry, especially circle equations and gradients of lines, before teaching this topic. Encourage pupils to draw diagrams to visualize the problems. Emphasize the two main methods for finding the equation of a tangent (direct formula and differentiation) and when to apply each. Provide ample practice problems for both tangent equations and length of tangents.