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 2, Routine Procedures

In the diagram, a circle having centre M touches the x-axis at A(−1 ; 0) and the y-axis at B(0 ; 1). 4.1 Determine the equation of the circle centred at M in the form (x − a)² + (y − b)² = r².

Source: DBE Mathematics Paper 2, November 2019, Question 4.1

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

4 M(−3 ; 4) is the centre of the large circle and a point on the small circle having centre O(0 ; 0). NM is a tangent to the smaller circle at M, with MOS a diameter. 4.3 Determine the equation of NM in the form y = mx + c.

Source: DBE Mathematics Paper 2, November 2020, Question 4.3

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.