3 Fig. 3 shows an ellipse with parametric equations \(x = a \cos \theta , y = b \sin \theta\), for \(0 \leqslant \theta \leqslant 2 \pi\), where \(0 < b \leqslant a\).
The curve meets the positive \(x\)-axis at A and the positive \(y\)-axis at B .
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\caption{Fig. 3}
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- Show that the radius of curvature at A is \(\frac { b ^ { 2 } } { a }\) and find the corresponding centre of curvature.
- Write down the radius of curvature and the centre of curvature at B .
- Find the relationship between \(a\) and \(b\) if the radius of curvature at B is equal to the radius of curvature at A . What does this mean geometrically?
- Show that the arc length from A to B can be expressed as
$$b \int _ { 0 } ^ { \frac { \pi } { 2 } } \sqrt { 1 + \lambda ^ { 2 } \sin ^ { 2 } \theta } d \theta$$
where \(\lambda ^ { 2 }\) is to be determined in terms of \(a\) and \(b\).
Evaluate this integral in the case \(a = b\) and comment on your answer. - Find the cartesian equation of the evolute of the ellipse.