2022 NYJC J1 CT Q4

2022 NYJC J1 CT Q4

7 marks
Mid Year Exam

The curve \(C\) has parametric equations

\(x=5\alpha \sec \theta \), \(y=3\alpha \tan \theta \), where \(-\frac{\pi }{2}<\theta <\frac{\pi }{2}\)

and \(\alpha \) is a positive constant.

  1. Find the cartesian equation of \(C\).[2]
  2. Sketch \(C\), giving the coordinates of any points of intersection with the axes and the equations of any asymptotes.[3]
  3. Determine the range of values of \(k\) such that the line \(ky=x\) meets the curve \(C\).[2]

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Answer:(i) \(\frac{x^2}{25\alpha^2} - \frac{y^2}{9\alpha^2} = 1\), (iii) \(k < -\frac{5}{3}\) or \(k > \frac{5}{3}\)

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