9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { - 2 x }\) and \(k\) is a constant greater than 1 .
The curve crosses the \(y\)-axis at P and has a turning point Q .
\begin{figure}[h]
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\caption{Fig. 9}
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- Find the \(y\)-coordinate of P in terms of \(k\).
- Show that the \(x\)-coordinate of Q is \(\frac { 1 } { 4 } \ln k\), and find the \(y\)-coordinate in its simplest form.
- Find, in terms of \(k\), the area of the region enclosed by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 2 } \ln k\). Give your answer in the form \(a k + b\).
The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \mathrm { f } \left( x + \frac { 1 } { 4 } \ln k \right)\).
- (A) Show that \(\mathrm { g } ( x ) = \sqrt { k } \left( \mathrm { e } ^ { 2 x } + \mathrm { e } ^ { - 2 x } \right)\).
(B) Hence show that \(\mathrm { g } ( x )\) is an even function.
(C) Deduce, with reasons, a geometrical property of the curve \(y = \mathrm { f } ( x )\).
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