1. Monash university decides to be establish a new shuttle bus stop to cover the area of three campuses, Caulfield, Clayton and Peninsula. It is located at the point minimising the sum of squares of distances to three campuses. In this question, you find coordinates of the stop given three campus locations A = (a1, a2), B = (b1, b2), C = (c1, C2). (a) Let dA(r,y), dB(I, y), de(r, y) denote the distances from the point (r, y) to A, B, C, [3] respectively. Show that the function f(x, y) := d'A(x, y) + d;(x, y) + do(x, y) has unique stationary point S. (b) Let SA, SB, SC denote the vectors from A, B, C to the point S found in part (a), [2] respectively. Compute SA + SB + 80. (c) For any point (x, y), denote by u(x, y) the vector from S to (x, y). Express the vectors [4] from A, B, C to (x, y) in terms of SA, SB, S and u(x, y). Show that f(x, y) = IsAll? + IsBll2 + 1/scl/2 + 3|/u(x, y)|12, where f(x, y) is the function from part (a) and || . || is the Euclidian norm. Use this formula to explain why S is the global minimum of f(x, y). (d) Let Ao be the middle of the side BC. Compute the vector mA from A to A, and show [3] that S lies on AAo. Similarly define Bo, Co and prove that S is the intersection point of AAo, BBo, CCo (medians)