A point \(P(x,y)\) moves so that \(OP^2=\lambda y\), where \(O\) is the origin and \(\lambda\) is a constant. Show that the locus of \(P\) is a circle. State the coordinates of the centre of the circle and the length of its radius.

Using the same axes, find the locus of a point \(Q(x',y')\) which moves so that \(QR^2=\lambda' y'\), where \(R\) is the fixed point \((\alpha,0)\) and \(\lambda'\) is a constant.

Could we sketch the point \(P(x,y)\) and the line \(OP\)?

What’s the formula for the distance between two points?

Could we rearrange this into the standard form for a circle?

If the two loci touch each other, and \(\alpha\ne 0\), prove that \(\alpha^2=\lambda\lambda'\).

Could we draw a picture of this situation?

What’s the distance between the two centres here?