CAIE P3 2017 November — Question 6

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2017
SessionNovember
TopicImplicit equations and differentiation

6 The equation of a curve is \(x ^ { 3 } y - 3 x y ^ { 3 } = 2 a ^ { 4 }\), where \(a\) is a non-zero constant.
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 x ^ { 2 } y - 3 y ^ { 3 } } { 9 x y ^ { 2 } - x ^ { 3 } }\).
  2. Hence show that there are only two points on the curve at which the tangent is parallel to the \(x\)-axis and find the coordinates of these points.