What is the locus of these points fixed to a rod?

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The ends \(A\) and \(B\) of a rod of fixed length move on the positive \(x\) and \(y\) axes respectively. \(C\) is a point on \(AB\), and \(CD\) is perpendicular to \(AB\), as shown in the figure. The lengths of \(AC\), \(BC\) and \(CD\) are \(a\), \(b\) and \(c\) respectively.

1. Find parametric equations for the locus of \(C\).
2. Find parametric equations for the locus of \(D\).
3. If \(c^2=ab\) show that the locus of \(D\) consists of part of a straight line and find the coordinates of the extremities of this locus.
4. Sketch the loci of \(C\) and \(D\) for the case \(a=4\), \(b=9\), \(c=6\).