What is covered in Grade 12 Mathematics Analytical Geometry?

Equation of a circle with centre not at origin. Equation of a tangent to a circle. Coordinate geometry problems combining line and circle.

Is Grade 12 Analytical Geometry tested in Paper 1 or Paper 2?

Grade 12 Mathematics Analytical Geometry is assessed in Paper 2 of the end-of-year examinations, as per the CAPS Mathematics curriculum guidelines.

When is Analytical Geometry taught in Grade 12 Mathematics?

Analytical Geometry is taught in Term 2 of Grade 12 according to the CAPS Annual Teaching Plan (ATP) for Mathematics in South Africa.

Is Grade 12 Mathematics Analytical Geometry on the CAPS curriculum?

Yes. Analytical Geometry is part of the official CAPS Mathematics curriculum for Grade 12, Term 2 in South Africa. The content on MathSciBuddy follows the Annual Teaching Plan (ATP) set by the Department of Basic Education.

Is this Grade 12 Mathematics study material free?

Yes. All chapter notes, definitions, formulas, and worked examples on MathSciBuddy are completely free to read. No account or subscription is needed to access CAPS-aligned study notes.

Grade 12 Mathematics

Grade 12 · Term 2Mathematics

Analytical Geometry

We extend circle geometry on the Cartesian plane to include finding tangent and chord properties algebraically, and applying gradient and line equations in circle problems.

Week 3

5.1 Circles, Tangents & Chord Properties

Work with circles in completed-square form
Find equations of tangent lines to circles
Apply perpendicular bisector of a chord passes through centre
Solve problems combining circle and line equations

🌍

Real-World Connection

Radar systems detect aircraft by finding intersections of circles (range rings) from different stations. The analytical geometry of circles and their tangents is the mathematical backbone of triangulation.

Property / Rule

Perpendicular from Centre to Chord

The perpendicular from the centre of a circle to a chord bisects the chord. Conversely, the perpendicular bisector of any chord passes through the centre.

OC\perp AB\Rightarrow AC=CB

Property / Rule

Finding the Centre from Three Points

The perpendicular bisectors of any two chords intersect at the centre. Set up equations of two perpendicular bisectors and solve simultaneously.

\text{Centre} = \text{intersection of }\ \perp\text{-bisectors}

💡 Tip

A tangent meets a circle at EXACTLY one point. To find where a line meets a circle, substitute the line equation into the circle equation. For a tangent, the resulting quadratic has discriminant $\Delta=0$ (one repeated solution).

Worked Examples

Worked Example

Chord and perpendicular bisector

Problem

x^2+y^2=25

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge1 mark

State the theorem about the perpendicular from the centre to a chord.

L1 · Knowledge3 marks

Find the centre of circle

x^2-4x+y^2+6y=12

by completing the square.

L2 · Routine Procedures4 marks

Find the equation of the tangent to

x^2+y^2=40

P(2,6)

L2 · Routine Procedures4 marks

Show that the line

y=3x-10

is tangent to the circle

x^2+y^2=10

L3 · Complex Procedures5 marks

Find the centre of the circle passing through

A(0,0)

B(4,0)

and

C(0,6)

L3 · Complex Procedures5 marks

Two circles intersect at

A(1,3)

and

B(-1,1)

. Both circles have centre on the

x

-axis. Find the centres.

L4 · Problem Solving6 marks

Find the length of the common chord of circles

x^2+y^2=25

and

(x-4)^2+y^2=9

L4 · Problem Solving5 marks

A circle has centre

(3,1)

and passes through origin. Find the equation and the tangent at origin.

Trigonometry

Calculus