The asteroid, a curve \(C\) which is used to characterize various properties of energy and magnetism, has parametric equations
where \(0 \le t \le \frac{1}{2}\pi\) and \(a\) is a positive constant.
Find the equation of the tangent to \(C\) at the point \(P\) with parameter \(p\).[3]
The tangent at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\). Show that the length \(AB\) depends only on \(a\).[3]
Find a cartesian equation of \(C\).[2]
The region bounded by \(C\) and the \(x\)- and \(y\)-axes is rotated through \(360^\circ\) about the \(y\)-axis. Find the exact value of the volume of revolution of the solid formed.[4]