Coordinate geometry

Coordinate geometry uses algebra to describe lines and circles. With coordinates you can find lengths, midpoints, gradients and equations, and decide whether lines are parallel or perpendicular.

Key facts

  • Gradient = (y₂ − y₁)/(x₂ − x₁).
  • Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).
  • Distance = √((x₂ − x₁)² + (y₂ − y₁)²).
  • Parallel lines: equal gradients. Perpendicular: gradients multiply to −1.
  • Circle centre (a, b), radius r: (x − a)² + (y − b)² = r².

Worked examples

Example 1

Find the equation of the line through (1, 3) perpendicular to y = 2x + 5.

  1. Gradient of the given line: 2. Perpendicular: −½.
  2. y − 3 = −½(x − 1)
  3. y = −½x + 3.5

Answer: y = −½x + 3.5

Example 2

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

  1. (x − 3)² − 9 + (y + 2)² − 4 − 3 = 0
  2. (x − 3)² + (y + 2)² = 16
  3. Centre (3, −2), radius 4.

Answer: Centre (3, −2), radius 4