Analytical Geometry

Analytical geometry uses coordinates and algebra to study points, lines, midpoints, gradients, distances and circles on the Cartesian plane. In Matric you'

Analytical geometry uses coordinates and algebra to study points, lines, midpoints, gradients, distances and circles on the Cartesian plane. In Matric you'll combine the distance, midpoint and gradient formulas with circle and line equations to solve multi-step problems.

Key concepts

  • Distance: d = √((x₂−x₁)² + (y₂−y₁)²)
  • Midpoint: M = ((x₁+x₂)/2 , (y₁+y₂)/2)
  • Gradient: m = (y₂−y₁)/(x₂−x₁)
  • Parallel lines: m₁ = m₂
  • Perpendicular lines: m₁·m₂ = −1
  • Equation of a line: y − y₁ = m(x − x₁)
  • Circle centred at (a,b): (x−a)² + (y−b)² = r²

How it works

Start by plotting the given points. Decide which formula the question actually wants: distance for lengths, midpoint for the middle of a segment, gradient when you need a slope, and the line/circle equation when you must describe a shape. Always check whether lines are parallel (same gradient) or perpendicular (gradients multiply to −1).

Step by step

1. Read carefully

Identify all given coordinates and what is being asked (length, gradient, equation, intersection).

2. Choose the formula

Match the unknown to its formula. Write the formula before substituting.

3. Substitute carefully

Use brackets around negatives. Double-check x₂−x₁ vs y₂−y₁ in the right order.

4. Simplify and check

Round only at the end (if asked). Verify by plotting roughly.

Worked examples

Find the equation of the line through A(2, 3) and B(6, 11).

  1. Gradient m = (11 − 3) / (6 − 2) = 8/4 = 2
  2. Use point-slope with A(2,3): y − 3 = 2(x − 2)
  3. Simplify: y = 2x − 1

Answer: y = 2x − 1

Show that AB is perpendicular to CD given A(1,2), B(3,6), C(0,4), D(4,2).

  1. m(AB) = (6−2)/(3−1) = 2
  2. m(CD) = (2−4)/(4−0) = −1/2
  3. m(AB)·m(CD) = 2 × −1/2 = −1 ✓

Answer: AB ⟂ CD

Find the equation of the circle with centre (2; −1) that passes through (5; 3).

  1. r² = (5 − 2)² + (3 + 1)² = 9 + 16 = 25
  2. Substitute the centre and r² into (x − a)² + (y − b)² = r²

Answer: (x − 2)² + (y + 1)² = 25

Common mistakes

  • Swapping x and y in the gradient formula
  • Forgetting the negative reciprocal rule for perpendicular lines
  • Expanding (x − a)² incorrectly when working with circles
  • Confusing midpoint with average of only one coordinate

Frequently asked questions

How much of Paper 2 is Analytical Geometry?

Typically 30–40 marks across one or two questions. It is one of the most predictable scoring opportunities in Paper 2.

Do I need to memorise the circle equation?

Yes — both (x−a)² + (y−b)² = r² and the expanded form x² + y² + Dx + Ey + F = 0 appear in past papers.

How do I find the angle of inclination?

Use tan θ = m. If the gradient is negative, add 180° to the calculator's angle to get an inclination between 90° and 180°.

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