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\includegraphics[max width=\textwidth, alt={}, center]{2734e846-f640-4203-ac11-6b2180a21950-4_337_944_255_557} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres are moving on a horizontal surface when they collide. Before the collision, \(A\) is moving with speed \(a \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\alpha\) with the line of centres and \(B\) is moving towards \(A\) with speed \(b \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\beta\) with the line of centres (see diagram). After the collision, \(A\) moves with velocity \(2 \mathrm {~ms} ^ { - 1 }\) in a direction perpendicular to the line of centres and \(B\) moves with velocity \(2 \mathrm {~ms} ^ { - 1 }\) in a direction making an angle of \(45 ^ { \circ }\) with the line of centres. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).

1. Show that \(a \cos \alpha = \frac { 5 } { 6 } \sqrt { 2 }\) and find \(b \cos \beta\).
2. Find the values of \(a\) and \(\alpha\).