A variable point \(P\) on the circle \(x^2+y^2=a^2\) has coordinates \((a\cos\theta,a\sin\theta)\), and \(A\) is the fixed point \((c,0)\).

If \(Q\) is the point on \(AP\) such that \(AQ:QP=\lambda:1\), find the coordinates of \(Q\), and show that the locus of \(Q\) is another circle. Calculate the coordinates of the centre of this circle and the length of its radius.

Show that the condition for the two circles to intersect at right angles is \(c^2=a^2(1+2\lambda+2\lambda^2)\).