A point $Y$ moves to define a point $P$ — can we find the locus of $P\,$?

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\(X\) is a fixed point \((3,0)\), \(Y\) is the point \((0,a)\). A line is drawn through \(Y\) and perpendicular to \(XY\), meeting the \(x\)-axis at \(Q\). On this line a point \(P\) is taken so that \(Y\) is the mid-point of \(QP\). Find the coordinates of \(P\).

Find the equation of the locus of \(P\) for varying values of \(a\).