Analytical geometry

Circles & tangents

Equation of a circle, tangent perpendicular to the radius, and the condition for a line to touch a circle.

CAPS

Grade 12 · Mathematics · Analytical geometry

This sits on Mathematics Paper 2.

The paper
Paper 2 is Euclidean geometry, analytical geometry, trigonometry and statistics. Diagrams are part of the question. A reason in a geometry proof must name the theorem, not only the result.
Cognitive level
CAPS spreads the marks across knowledge, routine procedures, complex procedures and problem solving. A routine question still wants the line of working. A problem-solving question wants you to choose the method, not only to calculate.
The relation
(x − a)²+(y − b)²=r². Circle
How a marker reads it
Prove the given point lies on the circle first — substitution of (6; 2) into the equation.

Each Mathematics paper is 150 marks. A correct substitution can earn a method mark even when the last arithmetic slips, if the formula was the right one.

Diagram

Worked example

Circle (x − 2)² + (y + 1)² = 25. Find the equation of the tangent at (6; 2).

Before any working — hunt first

Given

Numbers and units already in the sentence: 2 · 1 · 25 · 6 · 2. Copy each one into a column. A name without a number is not given.

Missing

After you copy the given, the letter that still has no number is missing. The paper is pointing at: the equation of the tangent at (6; 2). Circle that word — that is the missing variable.

Find

We are required to find: the equation of the tangent at (6; 2). Write that as a symbol on the left, with a question mark, before you touch a formula.

Calculate

Write (x − a)² + (y − b)² = r² with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives the equation of the tangent at (6; 2).

Simplified approach

Radius runs from centre (2; −1) to (6; 2). Tangent slope is the negative reciprocal.

Culture note

A circular traffic island in Polokwane — the kerb at one point is the tangent, the spoke to the fountain is the radius.

Formulas

(x − a)²+(y − b)²=r²

Circle

m_radius×m_tangent=−1

Tangent ⟂ radius

How to hunt the missing letter

Do not start by naming the letters. First hunt the missing one: copy every number you can see (Given). The letter with no number is Missing. The asking word is Find. The formula that still contains that letter is what you Calculate.

How to hunt: (1) Underline every number and its unit — those are Given. (2) Circle the asking word — find, calculate, how far, what is. That word names Find. (3) The letter the asking word names, which has no number yet, is Missing. (4) Write the formula with empty letters. Cross out every given letter. The letter still standing is what you Calculate.

a, b
centre (a; b)
How you spot this in the sentence
When the sentence names the centre (a; b), that letter is a, b. Copy the number next to it into Given. If the sentence never names it, a, b is Missing. Circle centre. Radius from centre to a point on the circle is perpendicular to the tangent.
Example:Centre (2; −1), point (2; 3): radius is vertical so tangent is horizontal.
r
radius
How you spot this in the sentence
When the sentence names the radius, that letter is r. Copy the number next to it into Given. If the sentence never names it, r is Missing. Distance from centre to any point on the circle.
Example:m_radius × m_tangent = −1 for a perpendicular pair.

Exam tricks you must know

These are the moves markers look for — not shortcuts that skip the method, but the habits that earn the method mark.

Always CAST — the second quadrant is a mark on its own
How:Sketch CAST. Read the sign of sin, cos or tan from the quadrant, not from memory of a special angle. For a general solution write the acute angle first, then 180° − θ or 360° − θ. Leaving the second angle is the KZN and Gauteng dock.
When to use it:Any sin / cos / tan equation that asks for θ in an interval.
Decide first: is money going in (F) or a loan being paid off (P)?
How:Underline the asking word. “Future value”, “how much in the account”, “accumulate” → F. “Present value”, “loan”, “outstanding balance”, “how much can she borrow” → P. Write F or P in the Find cell before you open the formula sheet. The wrong letter loses the whole question.
When to use it:Every compound-interest and annuity question on Paper 1.
Ask: does ABC differ from ACB? Yes → permutation. No → combination
How:Read the noun. Arrangements, passwords, in a line, first/second/third → ⁿPᵣ. Committees, teams, groups, chosen → ⁿCᵣ. Write that sentence as the method line. Then substitute n and r.
When to use it:Counting, arrangements, and probability “how many ways” questions.

Step 1 / 6

Radius slope

m_r=(2 − (−1))(6 − 2)=34.

m_r=34

Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.

4. Step-by-step working

01
Radius slope
m_r=(2 − (−1))(6 − 2)=34.
m_r=34
02
Tangent slope
m_t=−43.
m_t=−43
03
Equation
y−2=(−4/3)(x − 6)→4x+3y−30=0.
4x+3y−30=0
04
Step F — Box the answer with its unit
4x+3y−30=0

More worked examples

Board walkthroughs on this topic. Open one, copy the working, then try the next.

Example 1 / 20

A line has run 1 and rise 1. Gradient and inclination θ (to the nearest degree)?

4. Step-by-step working

01
Gradient
m=11=1.
m=1
02
Calculate
θ=tan⁻¹(1) ≈ 45°.
θ ≈ 45°

How we got there

Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.

Explain in another language · 11

1. What is the question asking?

Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: A line has run 1 and rise 1. Gradient and inclination θ (to the nearest degree)?

In other words — as if you have never seen this before

If you have never seen this before: m = Δy/Δx. θ = tan⁻¹ m (inclination to the positive x-axis). We will name a method in words, write the formula with no numbers, then substitute one line at a time.

2. What do we already know?

Δx=1, Δy=1.

3. Which method — and why this one?

m = Δy/Δx. θ = tan⁻¹ m (inclination to the positive x-axis).

4. Step-by-step working

01 / 05
Gradient
m=11=1.
m=1
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
02 / 05
Calculate
θ=tan⁻¹(1) ≈ 45°.
θ ≈ 45°
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
m=1, θ ≈ 45°
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.

5. The answer

m=1, θ ≈ 45°

Copy onto your answer sheet — in this order

Formula: m=1

Answer: m=1, θ ≈ 45°

6. Check it like a marker

Inclination is an angle, not the gradient itself. Take tan⁻¹.

7. A trap first-timers fall into

First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”

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