Intersection Problems

This lesson covers finding intersection points between lines and curves, between two curves, and using parametric equations in intersection problems. These techniques are essential for the Edexcel 9MA0 specification and combine algebra with coordinate geometry.

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Line Meets Curve

To find the intersection of a line and a curve, substitute the equation of the line into the equation of the curve (or vice versa) and solve the resulting equation.

Example: Line and Parabola

Find where y = 2x − 1 meets y = x² − 2x + 3.

Set equal: 2x − 1 = x² − 2x + 3 0 = x² − 4x + 4 0 = (x − 2)² x = 2 (repeated root)

When x = 2: y = 3. The line is tangent to the parabola at (2, 3).

Example: Line and Circle

Find where 2x + y = 10 meets x² + y² = 50.

From the line: y = 10 − 2x. Substitute: x² + (10 − 2x)² = 50 x² + 100 − 40x + 4x² = 50 5x² − 40x + 50 = 0 x² − 8x + 10 = 0