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

The red circle here is \(x^2 + y^2 = \lambda y\), while the green circle is \((x-\alpha)^2+y^2=\lambda'y\).

What happens as we vary \(\lambda, \alpha\) and \(\lambda'\)? When do the circles touch?