Introduction
Every point where two graphs cross or touch corresponds exactly to a value of (and ) that satisfies both equations simultaneously. This connection between geometry and algebra is fundamental throughout Pure Mathematics 1: it underpins curve sketching, optimisation, and — most directly tested in 9709 — questions asking you to find or classify intersections of a line with a quadratic curve.
A classic exam task reads: "Find the set of values of for which the line intersects, touches, or does not meet the curve ." Solving this requires translating a geometric question into an algebraic one via the discriminant.
Core Concept
From geometry to algebra
If two curves have equations and , their intersection points satisfy both equations at once. Setting produces a single equation whose solutions give the -coordinates of the intersection points. The corresponding -values are found by substituting back.
| Number of real solutions of | Geometric interpretation |
|---|---|
| Two distinct real solutions | Line/curve cuts the other curve at two points |
| One repeated real solution | Line/curve touches (is tangent to) the other curve |
| No real solutions | Line/curve does not meet the other curve |
Using the discriminant
When one graph is a straight line and the other is a quadratic curve, eliminating produces a quadratic equation in :
The nature of the intersections is then fully determined by the discriminant :
| Condition | Nature of intersection |
|---|---|
| Two distinct intersection points | |
| Exactly one point of contact (tangency) | |
| No intersection |
This is the key algebraic tool for the syllabus objective.
Unlock the full Coordinate Geometry note with Nova
You're reading the preview. Unlock the complete note — every worked example, examiner pitfall and practice question — plus 24/7 AI tutoring from Nova that teaches directly from these notes.