1. Find equation of a tangent using perpendicular and y=mx + c
Grade 9
Equation of a tangent
Find equation of a tangent using perpendicular and y=mx + c
If you have any questions regarding these resources or come across any errors, please contact

2. Lesson Plan Lesson Overview Progression of Learning
Objective(s)
Find equation of a tangent using perpendicular and y=mx + c
Grade 9
Prior Knowledge
Surds
Circle vocabulary
Circle theorems
Equation of a circle (lesson to be completed prior to this lesson)
y = mx + c
Duration
40 minutes.
Resources
Print slides:
Equipment
Progression of Learning
What are the students learning?
How are the students learning? (Activities & Differentiation)
Recap key circle terminology and the radius meets tangent circle theorem
Using slide 3 review the key words associated with circles and the circle theorem relevant to this lesson. 5
Linking equation of a circle with equation of a tangent
Using slides 4 and 5 explain how to find the equation of a tangent when the gradient is 0.
Finding the equation of a tangent
Give students slide 13 printed. Using slide 6 go through each step in order to find the equation of a tangent. Will need to relate to prior learning about gradients and y – mx + c. Further question given for practice. Solution to practice question on slide 8. 15
Finding the equation of a tangent in contextualised problems
Give students slide 14 printed. Students to work independently on the question before reviewing collectively.
Finding the equation of a tangent in exam questions (from specimen papers)
Give students slide 15 This includes 3 exam questions related to objective. Students need to use notes from lesson to answer the questions. Ensure that all steps are shown. Relate to mark scheme to show how the marks are allocated. 10
Next Steps
Assessment
PLC/Reformed Specification/Target 9/Algebra/Equation of a tangent

3. Tangent meets radius at 90°
Key Vocabulary
Equation Circle Origin Radius Diameter Centre Tangent
Tangent meets radius at 90°

4. Recap - The equation of a circle
The equation of this circle with centre (0, 0) is x2 + y2 = r2.
radius
Centre (0,0)
x axis
y axis

5. The equation of a tangent - 1
There are two tangents to this circle with equation
x2 + y2 = 42 with a gradient of 0. 4
tangents
radius
x axis 4
Centre (0,0) -4
y axis
The equations of these tangents are y = 4 and y = -4

6. The equation of a tangent - 2
Write down the equation of a tangent to a circle with centre C(0, 0) and radius 5 that goes through the point A(3,4).
1: Find gradient of AC
= difference in y
difference in x
4 3
C(0,0) 5
2: Find gradient of tangent
Because radius (AC) and tangent are perpendicular:
A (3,4)
−3 4
gradient of radius X gradient of tangent = -1
3: Equation of tangent using y = mx + c
Sub in coordinate (3,4)
Find c
Equation
𝑦=− 3 4 𝑥+𝑐
4=− 3 4 x 3+𝑐
16 4 =− 𝑐
𝑦=− 3 4 𝑥+ 25 4
4=− 𝑐
25 4 =𝑐

7. The equation of a tangent – 2 (practice)
Write down the equation of a tangent to a circle with the equation x2 + y2 = 52 that goes through the point A(4,6).

8. The equation of a tangent – 2 (solution)
Write down the equation of a tangent to a circle with the equation x2 + y2 = 52 that goes through the point A(4,6).
Gradient of AC = =
Gradient of tangent =
y = x + c = (4) + c c =
y = x +

9. Problem Solving and Reasoning
The line l1 is a tangent to a circle with the equation x2 + y2 = 52 at the point P.
P is the point (4, -6).
The line l1 crosses the x–axis at the point Q and the y-axis at the point S.
Work out the area of triangle OSQ.
Gradient of tangent = 2/3
y = 2/3x + c using (4,-6) y = 2/3x – 26/3
Tangent crosses y-axis when x = 0 leading to (0,-26/3)
Tangent crosses x-axis when y = 0 leading to (13,0)
Area of triangle OPQ = ½ x base x perpendicular height
= ½ x 26/3 x 13
= 56 1/3 units squared

10. Exam Questions – Specimen Papers
5 marks

11. Exam Questions – Specimen Papers
5 marks

12. Exam Questions – Specimen Papers

13. The equation of a tangent - 2
Write down the equation of a tangent to a circle with centre C(0, 0) and radius 5 that goes through the point A(3,4).
1: Find gradient of AC
PRACTICE
Write down the equation of a tangent to a circle with the equation x2 + y2 = 52 that goes through the point A(4,6).
2: Find gradient of tangent
3: Equation of tangent using y = mx + c
Student Sheet 1

14. Problem Solving and Reasoning
The line l1 is a tangent to a circle with the equation x2 + y2 = 52 at the point P.
P is the point (4, -6).
The line l1 crosses the x–axis at the point Q and the y-axis at the point S.
Work out the area of triangle OSQ.
Student Sheet 2

15. Exam Questions – Specimen Papers
5 marks
5 marks
Student Sheet 3