A line and a curve

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What this lesson covers

  • Substitute the line into the curve.
  • Solve the quadratic for x, then find y.
  • A discriminant of zero means the line is a tangent.
Worked example

Two crossings

  • y = x + 1 and y = x² − 1.
  1. x² − 1 = x + 1
  2. x² − x − 2 = 0
  3. (x − 2)(x + 1) = 0
  4. points (2, 3) and (−1, 0)
Worked example

A tangent

  • Find k so that y = 2x + k touches y = x².
  1. x² − 2x − k = 0
  2. b² − 4ac = 4 + 4k
  3. 4 + 4k = 0one touching point
  4. k = −1

Common mistakes

  • Giving only the x values.
    Each intersection is a point, so give y too.
  • Substituting x into the curve and forgetting the line check.
    Put your point into both equations.

Your turn

  • 1. Solve y = 2x and y = x² − 3.
Show answer

1. (3, 6) and (−1, −2)

Summary

What to take away

  • Substitute to get one quadratic.
  • The discriminant counts the meeting points.

More in AS 91269

Simultaneous linear equationsNotesA line and a curveNotesWhat the solutions meanNotesModelling with systemsNotes

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