One distance is double another; can we find a locus?

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Given that \(A\) is the point \((0,3)\) and \(B\) is the point \((0,-3)\), a point \(P(x,y)\) moves so that \(PA=2PB\). Show that the equation of the locus of \(P\) is \[ x^2 + y^2 +10y + 9 = 0.\]

How can we work out the distances \(PA\) and \(PB\) (in terms of \(x\) and \(y\))?

In this applet, changing \(\theta\) will move \(P\), such that \(PA = 2PB\). What shape is the locus of \(P\)?