In the \(x\)–\(y\) plane, the point \(A\) has coordinates \((a,0)\) and the point \(B\) has coordinates \((0,b)\), where \(a\) and \(b\) are positive. The point \(P\), which is distinct from \(A\) and \(B\), has coordinates \((s,t)\). \(X\) and \(Y\) are the feet of the perpendiculars from \(P\) to the \(x\)-axis and \(y\)-axis respectively, and \(N\) is the foot of the perpendicular from \(P\) to the line \(AB\).
Use this applet to explore the behaviour as \(A\), \(B\) and \(P\) are moved.
Show that, if \[\left(\frac{t - b}{s}\right)\left(\frac{t}{s - a}\right) = -1,\] then \(N\) lies on the line \(XY\).
Give a geometrical interpretation of this result.
What can you say about the position of \(P\) when \(N\) is on \(XY\)?
You can show the circle that has \(AB\) as a diameter.
