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Quadratics – Intersection of a line and a quadratic curve

📘 Notes

Intersection of a Line and a Quadratic (Year 12)

To find where a straight line meets a quadratic curve, we form a quadratic equation by substitution.

1. Method Summary

  1. Write both equations in the form \(y = \dots\) or \(x = \dots\).
  2. Substitute the linear equation into the quadratic.
  3. Solve the resulting quadratic equation.
  4. Substitute back to find the corresponding values of the other variable.
Solutions give the coordinates of intersection points.

2. Example 1 — Basic Substitution

Find the points where

\[ y = x + 1 \quad \text{and} \quad y = x^2 - 4 \]

Step 1: Substitute \(y = x + 1\) into \(y = x^2 - 4\):

\[ x + 1 = x^2 - 4 \Rightarrow x^2 - x - 5 = 0 \]

Step 2: Solve the quadratic:

\[ x = \frac{1 \pm \sqrt{1 + 20}}{2} = \frac{1 \pm \sqrt{21}}{2} \]

Step 3: Substitute into \(y = x + 1\):

\[ y = \frac{1 \pm \sqrt{21}}{2} + 1 = \frac{3 \pm \sqrt{21}}{2} \]

Intersection points: \[ \left(\frac{1+\sqrt{21}}{2},\,\frac{3+\sqrt{21}}{2}\right) \quad \text{and} \quad \left(\frac{1-\sqrt{21}}{2},\,\frac{3-\sqrt{21}}{2}\right) \]

3. Example 2 — Using the Discriminant

How many intersection points do the curves have?

\[ y = 2x + 3, \quad y = x^2 - x + 1 \]

Substitute into the quadratic:

\[ 2x + 3 = x^2 - x + 1 \Rightarrow x^2 - 3x - 2 = 0 \]

Use the discriminant \( \Delta = b^2 - 4ac \):

\[ \Delta = (-3)^2 - 4(1)(-2) = 9 + 8 = 17 > 0 \]

Two intersection points (the line crosses the curve twice).

4. Example 3 — No Intersection

Determine whether the line meets the curve:

\[ y = 4x + 1, \quad y = x^2 - 2x + 5 \]

Substitute:

\[ 4x + 1 = x^2 - 2x + 5 \Rightarrow x^2 - 6x + 4 = 0 \]

Discriminant:

\[ \Delta = (-6)^2 - 4(1)(4) = 36 - 16 = 20 > 0 \]

Two intersection points (line cuts the curve in two places).

5. Example 4 — Tangent Line (One Intersection)

Does the line touch the curve?

\[ y = 3x + 4, \quad y = x^2 + 6x + 11 \]

Substitute:

\[ 3x + 4 = x^2 + 6x + 11 \Rightarrow x^2 + 3x + 7 = 0 \]

Discriminant:

\[ \Delta = 3^2 - 4(1)(7) = 9 - 28 = -19 < 0 \]

No real intersection (line does not touch or cross the curve).

6. Tips for Year 12

  • Always substitute into the quadratic, not the line.
  • Use the discriminant to check how many intersections:
\[ \Delta > 0: \text{two intersections} \qquad \Delta = 0: \text{one intersection (tangent)} \qquad \Delta < 0: \text{no intersection} \]
  • Solutions must be written in coordinate form \((x,y)\).

7. Try These

  1. \(y = 5x - 1,\quad y = x^2 + 3x - 2\)
  2. \(y = 2x + 6,\quad y = x^2 + 4x + 9\)
  3. \(x + y = 3,\quad y = x^2 - x + 1\)

Find the intersection points or show that there are none.