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Coordinate Geometry

14 min read

Angle between two lines

If two lines have gradients m₁ and m₂, the angle θ between them satisfies:

If m₁m₂ = −1 the lines are perpendicular; if m₁ = m₂ they are parallel.
Worked exampleAngle between lines

Find the angle between y = 2x and y = x/3.

  1. Gradients are 2 and 1/3.

Answer: θ = 45°.

Perpendicular distance

The shortest distance from a point (x₁, y₁) to the line ax + by + c = 0 is:

The perpendicular distance.
Worked exampleDistance from a point to a line

Find the distance from (2, 3) to the line 3x + 4y − 5 = 0.

  1. Substitute.

Answer: 2.6 units.

Locus of a moving point

A locus is the path of a point that moves according to a rule. The locus of a point at a fixed distance r from a point (a, b) is a circle: (x − a)² + (y − b)² = r². The locus of points equidistant from two points is their perpendicular bisector. The locus of points equidistant from a fixed point and a fixed line is a parabola.

Parts of a circleA circle with its parts labelled: the centre, the radius from the centre to the edge, the diameter through the centre, a chord joining two points on the circle, an arc which is part of the circumference, a sector between two radii, a segment cut off by a chord, and a tangent touching the circle at one point.centreradiusdiametersectorchordarcsegmenttangent

The ratio theorem

The point P that divides the line AB internally in the ratio m : n has coordinates:

A(x₁, y₁), B(x₂, y₂). For m = n this is the midpoint. For an external division, the ratio is taken as negative.
The locus of a point 5 units from the origin, shown as y = √(25 − x²).
Try itDistance to a line

The line 3x + 4y − 5 = 0. Choose a point.

Check yourself Lines of gradient 3 and −1/3 are
Check yourself The distance from the origin to the line 3x + 4y − 10 = 0 is
Check yourself The point dividing A(0, 0) and B(9, 6) in the ratio 1 : 2 is

Key terms

Locus
The path of a point moving by a rule.
Perpendicular distance
The shortest distance from a point to a line.
Internal division
A point between A and B in a given ratio.
Perpendicular bisector
The line perpendicular to AB through its midpoint.

Summary

  • tan θ = |(m₁ − m₂)/(1 + m₁m₂)|.
  • Distance from (x₁, y₁) to ax + by + c = 0 is |ax₁ + by₁ + c| ÷ √(a² + b²).
  • A locus is described by an equation.
  • The ratio theorem finds a dividing point.
Linear Programming Trigonometry