Given that \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\), show that \(1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \left( \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } x } + \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 2 } x } \right) ^ { 2 }\).
The arc of the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) for \(0 \leqslant x \leqslant \ln a\) (where \(a > 1\) ) is denoted by \(C\).
Show that the length of \(C\) is \(\frac { a - 1 } { \sqrt { a } }\).
Find the area of the surface formed when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
An ellipse has parametric equations \(x = 2 \cos \theta , y = \sin \theta\) for \(0 \leqslant \theta < 2 \pi\).
Show that the normal to the ellipse at the point with parameter \(\theta\) has equation
$$y = 2 x \tan \theta - 3 \sin \theta$$
Find parametric equations for the evolute of the ellipse, and show that the evolute has cartesian equation
$$( 2 x ) ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } = 3 ^ { \frac { 2 } { 3 } }$$
Using the evolute found in part (ii), or otherwise, find the radius of curvature of the ellipse
(A) at the point \(( 2,0 )\),
(B) at the point \(( 0,1 )\).