Grade 12 Analytical Geometry — How to Identify the Question Type

Grade 12 analytical geometry extends Grade 10 to include the equation of a circle and the equation of a tangent to a circle. The gradient relationship between a radius and its tangent (perpendicular) is central to most problems.

Type 1: Equation of a circle

Trigger words: "Equation of the circle", "centre and radius", "(x − a)² + (y − b)² = r²"

Trigger structure: A circle is defined by its centre and radius, or by an equation in the form x² + y² + Dx + Ey + F = 0.

Do not confuse with: Equation of a straight line — a circle equation always has both x² and y² with equal coefficients.

Method (no numbers — just the steps)

Standard form: (x − a)² + (y − b)² = r² where (a ; b) is the centre
To find centre and radius from x² + y² + Dx + Ey + F = 0: complete the square for both x and y
To find the equation: substitute the centre and radius directly into the standard form

Grade 12Paper 2Level 1 — Knowledge

3.1 A circle has centre (2 ; −3) and radius 5. Write down the equation of the circle.

Source: DBE Mathematics Paper 2, November 2022, Question 3.1

Common mistake

Reading the centre signs incorrectly — in (x − 3)² + (y + 2)² = 25 the centre is (3 ; −2), not (−3 ; 2).

See the progression — same type, increasing difficulty

Easy

Write the equation of the circle with centre (0 ; 0) and radius 4.

Practice question — not sourced from a past paper.

Medium

Find the centre and radius of the circle x² + y² − 6x + 4y − 3 = 0.

Practice question — not sourced from a past paper.

Hard

A circle passes through (0 ; 0), (4 ; 0), and (0 ; 6). Determine the equation of the circle.

Practice question — not sourced from a past paper.

Type 2: Equation of the tangent to a circle

Trigger words: "Tangent at the point", "equation of the tangent", "line that touches the circle"

Trigger structure: A point on the circle is given. The question asks for the equation of the line that touches the circle at that point.

Do not confuse with: Equation of the chord — a chord passes THROUGH the circle; a tangent only TOUCHES it at one point.

Method (no numbers — just the steps)

Find the gradient of the radius from the centre to the given point: m_r = (y − b)/(x − a)
The tangent is perpendicular to the radius: m_t = −1/m_r
Use y − y₁ = m_t(x − x₁) with the point of tangency as (x₁ ; y₁)

Grade 12Paper 2Level 3 — Complex Procedures

3.3 The circle has equation x² + y² = 25. The tangent touches the circle at (3 ; 4). Determine the equation of the tangent.

Source: DBE Mathematics Paper 2, November 2023, Question 3.3

See the progression — same type, increasing difficulty

Easy

The circle x² + y² = 50 has a tangent at (5 ; 5). Find the gradient of the tangent.

Practice question — not sourced from a past paper.

Medium

A circle has centre (1 ; 2) and the tangent touches at (4 ; 6). Find the equation of the tangent.

Practice question — not sourced from a past paper.

Hard

The circle (x − 2)² + (y − 1)² = 20 has tangents drawn from the external point (6 ; 5). Verify that (6 ; 5) lies outside the circle and determine the equation of one tangent.

Practice question — not sourced from a past paper.

Words like determine and hence appear throughout this topic — see the instruction word glossary for full definitions.