Grade 12 Analytical Geometry: Lines, Circles and Angles of Inclination

The core formulas

• Distance: d = √[(x₂ − x₁)² + (y₂ − y₁)²].
• Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).
• Gradient: m = (y₂ − y₁)/(x₂ − x₁).
• Parallel lines have equal gradients; perpendicular gradients multiply to −1.
• Angle of inclination: tan θ = m, with θ measured anticlockwise from the positive x-axis.
• Circle centred at (a, b): (x − a)² + (y − b)² = r².

Worked example: angle of inclination

A line has gradient m = −1. Then tan θ = −1, so the reference angle is 45° and θ = 180° − 45° = 135°.

If the gradient is negative, the angle of inclination is obtuse. Write that step out; it is a frequent mark.

Worked example: tangent to a circle

The circle x² + y² − 4x + 6y − 12 = 0 can be written by completing the square as (x − 2)² + (y + 3)² = 25, so the centre is (2, −3) and the radius is 5.

At the point (6, 0) on the circle, the radius has gradient (0 − (−3))/(6 − 2) = 3/4, so the tangent gradient is −4/3 and the tangent is y − 0 = −4/3(x − 6).

Proving shapes

• Parallelogram: both pairs of opposite sides have equal gradients, or the diagonals share a midpoint.
• Rhombus: a parallelogram with two adjacent sides equal in length.
• Rectangle: a parallelogram with one right angle, shown by m₁ × m₂ = −1.
• Collinear points: the gradient between each pair is the same.