Analytical Geometry for Matric Maths

Distance, gradient, midpoint, equation of a line and the circle for NSC Paper 2. CAPS-aligned with AI step-by-step explanations.

Analytical Geometry in Paper 2 builds on Grade 10–11 work: distance, gradient and midpoint formulas, equation of a straight line, parallel and perpendicular lines, and the equation of a circle (x − a)² + (y − b)² = r². Grade 12 adds the circle: finding its centre and radius by completing the square, writing the equation of a tangent at a given point, and deciding whether a point lies inside, on or outside a circle. Questions build step by step on one diagram, so an early answer such as a gradient or a centre is often reused later.

Key concepts

  • Distance formula d = √((x₂ − x₁)² + (y₂ − y₁)²)
  • Gradient m = (y₂ − y₁) / (x₂ − x₁)
  • Equation of a line in y = mx + c and point-gradient form
  • Equation of a circle and tangents to a circle
  • Angle of inclination: tan θ = m
  • Midpoint M((x₁ + x₂)/2, (y₁ + y₂)/2)
  • A tangent is perpendicular to the radius at the point of contact
  • Completing the square to find a circle's centre and radius
  • Collinear points have equal gradients

Worked examples

Find the gradient between A(2, 3) and B(6, 11).

m = (11 − 3) / (6 − 2) = 8/4 = 2.

Find the centre and radius of x² + y² − 6x + 4y − 12 = 0.

Complete the square: (x − 3)² − 9 + (y + 2)² − 4 − 12 = 0, so (x − 3)² + (y + 2)² = 25. Centre (3, −2) and radius 5.

Find the equation of the tangent to x² + y² = 25 at P(3, 4).

The radius from (0, 0) to P has gradient 4/3, so the tangent's gradient is −3/4. y − 4 = −3/4(x − 3), so y = −3/4 x + 25/4.

Find the inclination of the line y = −x + 2.

tan θ = −1. The reference angle is 45°, and a negative gradient means an obtuse inclination: θ = 180° − 45° = 135°.

Show that A(1, 2), B(3, 6) and C(5, 10) are collinear.

m_AB = (6 − 2)/(3 − 1) = 2 and m_BC = (10 − 6)/(5 − 3) = 2. The gradients are equal and B is a common point, so A, B and C are collinear.

Common mistakes

  • Forgetting that perpendicular gradients multiply to −1.
  • Mis-reading the centre of the circle when the equation isn't in standard form.
  • Giving a negative inclination: when tan θ is negative, θ = 180° minus the reference angle.
  • Using the gradient of the radius as the gradient of the tangent.

Exam tips

  • Always sketch the points before computing anything — it catches sign mistakes.
  • Write m₁ × m₂ = −1 explicitly when you use perpendicular lines; it is often a separate mark.
  • Compare the distance from the centre to a point with the radius to decide whether the point is inside, on or outside the circle.

In past papers

  • NSC Nov 2024 Paper 2 Q3 — circle and tangent

Frequently asked questions

Is the equation of an ellipse examined?

No. Only the equation of a circle is examined in Matric Maths.

How do I find the equation of a tangent to a circle?

Find the gradient of the radius to the point of contact, take its negative reciprocal as the tangent's gradient, then substitute the point into y − y₁ = m(x − x₁).

Which analytical geometry formulas are on the information sheet?

The distance, midpoint and gradient formulas, the straight-line equations, m = tan θ and the equation of a circle are provided. You still need to recognise when to use each one.

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