1.3 Coordinate geometry

Cambridge International AS & A Level Mathematics 9709 (2026-2027) · Paper 1: Pure Mathematics 1 · official syllabus

COORDINATE GEOMETRY LAB

Learn it by playing it

Four short missions on one live grid. When a mission glows green, read why it worked, then take the next one.

1234
Mission 1 of 4

Steer the gradient

A is pinned at (-2, -1). Drag B until the line through them climbs exactly 2 units for every 1 it runs - gradient exactly 2.

gradient = 0.50 - target 2 (rise 2.00 over run 4.00)

Study notes

Played the lab above? These notes are the reference version of what you just felt - use them to check exact formulas and the exam wording.

Straight lines 1.3

  • Gradient between two points: m = (y2 - y1) / (x2 - x1). Use any form of the equation: y = mx + c, y - y1 = m(x - x1), or ax + by + c = 0.
  • Given one point and the gradient, y - y1 = m(x - x1) is usually the fastest route; given two points, find m first.
  • Midpoint of AB: average the coordinates. Distance AB: Pythagoras on the coordinate differences.
  • Parallel lines have equal gradients; perpendicular gradients multiply to -1 (take the negative reciprocal).
  • To find where two lines meet, solve their equations simultaneously.

The circle equation 1.3

  • (x - a)² + (y - b)² = r² is the circle with centre (a, b) and radius r. Watch the signs: (x + 1)² means centre x = -1.
  • The expanded form x² + y² + 2gx + 2fy + c = 0 has centre (-g, -f) and radius sqrt(g² + f² - c). Complete the square to convert.

Lines and circles 1.3

  • A tangent is perpendicular to the radius at the point of contact: find the radius gradient, then take the negative reciprocal for the tangent.
  • The angle in a semicircle is a right angle, and a circle is symmetric about any diameter - both shorten geometry arguments.
  • Substitute the line equation into the circle equation to find intersections: the resulting quadratic has two, one or no solutions.
  • Implicit differentiation is NOT needed at P1 - tangents come from the radius property.

Graphs and equations 1.3

  • Points of intersection of two graphs are the solutions of their simultaneous equations - the picture and the algebra are the same fact.
  • For a line meeting a curve, substitute and inspect the discriminant: positive means two intersections, zero means the line touches (a tangent), negative means it misses.
  • Use this to find the set of values of a constant: for y = x + k against a quadratic, write the discriminant in terms of k and solve each case.

EduTribe study notes - original material aligned to the Cambridge International AS & A Level Mathematics 9709 syllabus (2026-2027). Automated checks completed; not yet reviewed by an educator.

Free public preview - no sign-in or pass needed. Automated checks completed; not yet reviewed by an educator. Sign in to save your practice progress across sessions.

Loading questions…