<p>The asteroid, a curve \(C\) which is used to characterize various properties of energy and magnetism, has parametric equations</p>

<p>where&nbsp;\(0 \le t \le \frac{1}{2}\pi\) and \(a\) is a positive constant.</p>
<ol class="abc-list" start="2"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Find the equation of the tangent to \(C\) at the point \(P\) with parameter \(p\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><div class="list-item-paragraph">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 \(a\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-2" data-list-item-path="imported-item-block-3-item-2"><div class="list-item-paragraph">Find a cartesian equation of \(C\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-3-item-3" data-list-item-path="imported-item-block-3-item-3"><div class="list-item-paragraph">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.<span class="inline-marks">[4]</span></div></li></ol>

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 \(p\).

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 \(a\).

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.