- The point \(A\) has coordinates \((5,16)\) and the point \(B\) has coordinates \((-4,4)\). The variable point \(P\) has coordinates \((x,y)\) and moves on a path such that \(AP = 2BP\). Show that the Cartesian equation of the path of \(P\) is \[(x + 7)^2 + y^2 = 100.\]

In this applet, as you move \(P\), the two bars show the relative sizes of \(AP\) and \(2\times BP\). When are they equal?

Can we find the lengths \(AP\) and \(BP\) in terms of \(x\) and \(y\)?

… Given that the path of \(Q\) is the same as the path of \(P\)…

What do we know about the equations of the paths of \(P\) and \(Q\) in this case? Must they be the same?