k is a positive real and P_{1}, P_{2}, ... , P_{n} are points in the plane. What is the locus of P such that ∑ PP_{i}^{2} = k? State in geometric terms the conditions on k for such points P to exist.

**Solution**

Let P_{i} have coordinates (a_{i}, b_{i}). Take the origin at the centroid of the points, so that ∑ a_{i} = ∑ b_{i} = 0. Then if P has coordinates (x, y), we have ∑ PP_{i}^{2} = ∑( (x - a_{i})^{2} + (y - b_{i})^{2} ) = n (x^{2} + y^{2}) + ∑(a_{i}^{2} + b_{i}^{2}). Let the origin be O, so that ∑(a_{i}^{2} + b_{i}^{2}) = ∑ OP_{i}^{2}. Then if ∑ OP_{i}^{2} > k, the locus is empty. Otherwise it is the circle with centre at the centroid of the n points and radius √( (k - ∑ OP_{i}^{2})/n ).

© John Scholes

jscholes@kalva.demon.co.uk

5 Mar 2002