Let P₁, P₂, P₃ and P₄ be the vertices of a square inscribed in a circle with center C(2, 3) . If the area of the square is 18 and the coordinates of P_i are (x_i, y_i) for i=1, 2, 3, 4 , then the value of _ i=1 ⁴ (x_i^2 + y_i^2) is equal to

Step-by-step solution

Let the origin be O(0, 0) . The square of the distance of the center C(2, 3) from the origin is OC^2 = 2^2 + 3^2 = 13 . The area of a square inscribed in a circle of radius r is given by 1 2 ( diagonal )^2 = 1 2 (2r)^2 = 2r^2 . Given that the area is 18 , we have 2r^2 = 18 r^2 = 9 . The required sum is _ i=1 ⁴ (x_i^2 + y_i^2) , which represents the sum of the squares of the distances of the vertices from the origin, i.e., _ i=1 ⁴ OP_i^2 . Let c be the position vector of the center C , and v _i be the vector from C