Intersection & Simultaneous Problems

This lesson focuses on finding where curves intersect — lines meeting curves, two curves meeting each other, and how the discriminant determines the nature of the intersection. These problems bring together algebra and coordinate geometry and are a staple of A-Level examinations.

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Line Meets Curve

To find where a straight line meets a curve, substitute the equation of the line into the equation of the curve. This produces an equation in one variable, which you solve.

Example 1: Find where the line y = 2x − 1 meets the parabola y = x².

Set equal: x² = 2x − 1
x² − 2x + 1 = 0
(x − 1)² = 0
x = 1 (repeated root)

y = 2(1) − 1 = 1

The line meets the parabola at exactly one point, (1, 1). The repeated root tells us the line is a tangent to the parabola.

Example 2: Find where y = x + 3 meets y = x² − x − 2.

x + 3 = x² − x − 2
x² − 2x − 5 = 0