Solve the equation
$$\sinh t + 7 \cosh t = 8$$
expressing your answer in exact logarithmic form.
A curve has equation \(y = \cosh 2 x + 7 \sinh 2 x\).
Using part (i), or otherwise, find, in an exact form, the coordinates of the points on the curve at which the gradient is 16 .
Show that there is no point on the curve at which the gradient is zero.
Sketch the curve.
Find, in an exact form, the positive value of \(a\) for which the area of the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = a\) is \(\frac { 1 } { 2 }\).