Coordinate geometry in the (x, y) plane (AS Unit 1: Pure Mathematics A) — WJEC A-Level Mathematics
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Coordinate geometry in the (x, y) plane (AS Unit 1: Pure Mathematics A) explained
The gradient m of a line through points (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁).
Read the full explanation
The point-slope form y − y₁ = m(x − x₁) directly generates the equation of a line passing through a known point with known gradient. The slope-intercept form y = mx + c displays the gradient m and y-intercept (0, c). The general linear form ax + by + c = 0 accommodates all lines, including vertical lines x = k where gradient is undefined. Two non-vertical lines with gradients m₁ and m₂ are parallel if and only if m₁ = m₂. They are perpendicular if and only if their gradients satisfy m₁m₂ = −1 (or m₂ = −1/m₁), which reflects the negative reciprocal relationship. Finding perpendicular bisectors combines the midpoint formula with perpendicular gradients.
Your focus
- Calculate gradients, midpoints, and lengths of line segments joining two points in the Cartesian plane.
- Form equations of straight lines in forms y = mx + c, y − y₁ = m(x − x₁), and ax + by + c = 0.
- Apply parallel (m₁ = m₂) and perpendicular (m₁m₂ = −1) gradient conditions to solve geometric problems.
Coordinate geometry in the (x, y) plane (AS Unit 1: Pure Mathematics A) exam tips
Marking Points
- calculating the gradient of a line through two given points
- applying the perpendicular gradient condition m₁m₂ = −1 to find a perpendicular slope
- forming the equation of the line using y − y₁ = m(x − x₁) and rearranging to specified form
Examiner Tips
- 💡Use y − y₁ = m(x − x₁) directly to avoid making substitution errors when finding c in y = mx + c.
- 💡Ensure equations are rearranged into the exact form requested, such as ax + by + c = 0 with integers.
Common Mistakes
- forgetting the negative sign when finding perpendicular gradients, writing m₂ = 1/m₁ instead of −1/m₁
- subtracting coordinates in inconsistent orders when calculating gradient (e.g. (y₂ − y₁)/(x₁ − x₂))