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Extended Only Coordinate Geometry

Line-Curve Intersections for IGCSE Maths

Finding where a straight line meets a curve by solving simultaneously. This subtopic is part of Coordinate Geometry in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding

What You Need to Know

Finding where a straight line meets a curve by solving simultaneously. This subtopic is part of Coordinate Geometry in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding line-curve intersections is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Line-Curve Intersections

The intersection of a straight line y = mx + c with a curve (usually a quadratic y = ax² + bx + c) is found by solving the simultaneous equations. Substitute the linear expression for y into the curve equation, then solve the resulting quadratic. The number of intersections is determined by the discriminant: two real intersections (b²−4ac > 0), one (tangent, b²−4ac = 0), or none (b²−4ac < 0). This is an Extended topic tested in Paper 4.

Step-by-Step Method

  1. 1

    Write both equations

    State the line y = mx + c and the curve (e.g. y = x² + 2x − 3) clearly.

  2. 2

    Substitute the linear expression into the curve

    Replace y in the curve equation with the expression from the line. This gives an equation in x only.

  3. 3

    Rearrange to standard form

    Bring everything to one side: ax² + bx + c = 0. Make sure the coefficient of x² is positive.

  4. 4

    Solve the quadratic

    Factorise, use the quadratic formula, or complete the square. Each x-value gives one intersection point.

  5. 5

    Find y-coordinates and state the points

    Substitute each x-value back into the linear equation (simpler) to find the corresponding y. Write the intersection points as coordinates.

Worked Example

Question

Find the coordinates of the points of intersection of the line y = 2x + 1 and the curve y = x² − x − 5.

Solution

Step 1: Set equal: x² − x − 5 = 2x + 1 Step 2: Rearrange. x² − x − 5 − 2x − 1 = 0 x² − 3x − 6 = 0 Step 3: Use the quadratic formula. x = (3 ± √(9 + 24))/2 = (3 ± √33)/2 x = (3 + 5.745)/2 ≈ 4.37 or x = (3 − 5.745)/2 ≈ −1.37 Step 4: Find y values using y = 2x + 1. For x ≈ 4.37: y ≈ 2(4.37) + 1 ≈ 9.74 For x ≈ −1.37: y ≈ 2(−1.37) + 1 ≈ −1.74 Answer: Intersection points ≈ (4.37, 9.74) and (−1.37, −1.74)

Exam Tips for Line-Curve Intersections

  • Always substitute the linear expression into the curve — not the other way round.
  • After substituting, collect all terms on one side before factorising or using the formula.
  • If the discriminant = 0, the line is a tangent to the curve — one repeated solution.
  • Find y by substituting back into the linear equation (not the quadratic) — it is quicker and less error-prone.

Practice Questions

Q1: Find the x-coordinates of intersection of y = x + 3 and y = x² − 2x + 1.

Show hint

Substitute: x² − 2x + 1 = x + 3 → x² − 3x − 2 = 0. Use the quadratic formula.

Q2: Show that the line y = 3x + 5 is tangent to the curve y = x² + x + 5.

Show hint

Substitute to get x² − 2x = 0. Find the discriminant: b² − 4ac = 4 − 0 = 4? Recheck: it factors to x(x−2) = 0 — two intersections, not tangent. Revise: for tangent, discriminant = 0.

Q3: Find the values of k for which the line y = x + k is tangent to y = x² − 4.

Show hint

Substitute: x² − x − 4 − k = 0. For tangency: discriminant = 1 + 4(4+k) = 0. Solve for k.

Frequently Asked Questions

What is line-curve intersections in IGCSE Maths?

Finding where a straight line meets a curve by solving simultaneously.

Is line-curve intersections in the Core or Extended syllabus?

Line-Curve Intersections is part of the Extended only syllabus for IGCSE Mathematics 0580.

How do I revise line-curve intersections effectively?

Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on line-curve intersections rather than trying to cover everything at once.

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