Find constant from gradient condition

5 The curve with equation $$6 \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { y } + \mathrm { e } ^ { 2 y } = c$$ where \(k\) and \(c\) are constants, passes through the point \(P\) with coordinates \(( \ln 3 , \ln 2 )\).

1. Show that \(58 + 2 k = c\).
2. Given also that the gradient of the curve at \(P\) is - 6 , find the values of \(k\) and \(c\).

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.} A curve has equation $$16 x ^ { 3 } - 9 k x ^ { 2 } y + 8 y ^ { 3 } = 875$$ where \(k\) is a constant.

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 k x y - 16 x ^ { 2 } } { 8 y ^ { 2 } - 3 k x ^ { 2 } }$$ Given that the curve has a turning point at \(x = \frac { 5 } { 2 }\)
2. find the value of \(k\)

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation $$y ^ { 3 } - x ^ { 2 } + 4 x ^ { 2 } y = k$$ where \(k\) is a positive constant greater than 1

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) lies on \(C\).
   Given that the normal to \(C\) at \(P\) has equation \(y = x\), as shown in Figure 2,
2. find the value of \(k\).

1. The curve \(C\) has equation

$$p x ^ { 3 } + q x y + 3 y ^ { 2 } = 26$$ where \(p\) and \(q\) are constants.

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a p x ^ { 2 } + b q y } { q x + c y }$$ where \(a\), \(b\) and \(c\) are integers to be found. Given that
   • the point \(P ( - 1 , - 4 )\) lies on \(C\)
   • the normal to \(C\) at \(P\) has equation \(19 x + 26 y + 123 = 0\)
   • find the value of \(p\) and the value of \(q\).