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Differentiation - Tangents and Normals at a Point

Differentiation - Tangents and Normals at a Point

TGM Original Questions

The point \(P\) lies on the curve \(y = 3x^3 - 2x + a\), where \(a\) is a constant. The equation of the tangent to the curve at \(P\) is \(9x + 9y - 43 = 0\). Find

  1. the coordinates of \(P\),
  2. the value of \(a\), and
  3. the equation of the normal to the curve at \(P\).

Solution:

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Answer:Two valid cases: (i) \(P\left(\dfrac13,\dfrac{40}{9}\right)\) or \(P\left(-\dfrac13,\dfrac{46}{9}\right)\) (ii) \(a=5\) or \(a=\dfrac{41}{9}\), respectively (iii) \(y=x+\dfrac{37}{9}\) or \(y=x+\dfrac{49}{9}\), respectively

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