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\includegraphics[max width=\textwidth, alt={}, center]{c342b622-a560-46da-9e64-edc4b7b3be93-4_606_915_219_557} The diagram shows two curves, \(C _ { 1 }\) and \(C _ { 2 }\), which intersect at the pole \(O\) and at the point \(P\). The polar equation of \(C _ { 1 }\) is \(r = \sqrt { 2 } \cos \theta\) and the polar equation of \(C _ { 2 }\) is \(r = \sqrt { 2 \sin 2 \theta }\). For both curves, \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). The value of \(\theta\) at \(P\) is \(\alpha\).

1. Show that \(\tan \alpha = \frac { 1 } { 2 }\).
2. Show that the area of the region common to \(C _ { 1 }\) and \(C _ { 2 }\), shaded in the diagram, is \(\frac { 1 } { 4 } \pi - \frac { 1 } { 2 } \alpha\).