Coordinate Geometry

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Find the equation of a straight line given sufficient information (e.g., given two points, or one point and the gradient)

  2. Interpret and use the different forms of line equations: y = mx + c, y – y₁ = m(x – x₁), and ax + by + c = 0. This includes calculating distances, gradients, midpoints, points of intersection, and understanding parallel and perpendicular lines

  3. Understand circle equations: (x – a)² + (y – b)² = r² represents a circle with centre (a, b) and radius r, including the expanded form x² + y² + 2gx + 2fy + c = 0

  4. Use algebraic methods to solve problems involving lines and circles, including geometric properties like tangent perpendicular to radius, angle in a semicircle, and symmetry

  5. Understand the relationship between a graph and its equation, and use intersections of graphs to solve equations (e.g., finding when a line intersects, touches, or misses a curve)

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1. Basic Formulas

When you have two points A(x₁, y₁) and B(x₂, y₂), you can find three important things:

Midpoint

The midpoint is the point exactly halfway between two points. Think of it as the average position.

Formula:

Midpoint=(x1+x22,y1+y22)\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

What it means: Add the x-coordinates and divide by 2, then add the y-coordinates and divide by 2.

Example: Find the midpoint of A(2, 5) and B(8, 11)

  • Midpoint = (2+82,5+112)=(5,8)\left( \frac{2 + 8}{2}, \frac{5 + 11}{2} \right) = (5, 8)

Distance

The distance is the straight-line length between two points. This comes from Pythagoras' theorem.

Formula:

Distance=(x2−x1)2+(y2−y1)2\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

What it means: Find the difference in x-values, square it. Find the difference in y-values, square it. Add them together and take the square root.

Example: Find the distance between A(1, 2) and B(4, 6)

  • Distance = (4−1)2+(6−2)2=9+16=25=5\sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 units

Gradient

The gradient (also called slope) measures how steep a line is. It tells you how much y changes when x increases by 1.

Formula:

Gradient(m)=y2−y1x2−x1\text{Gradient} (m) = \frac{y_2 - y_1}{x_2 - x_1}

What it means: The change in y divided by the change in x (rise over run).

Example: Find the gradient between A(1, 3) and B(5, 11)

  • Gradient = 11−35−1=84=2\frac{11-3}{5-1} = \frac{8}{4} = 2

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