In the plane rectangular coordinate system, two different moving points a and B on the parabola y = x ^ 2, which are different from the coordinate origin o, satisfy Ao ⊥ Bo (1) Find the trajectory equation of the center of gravity g of △ AOB (2) Is there a minimum value for the area of △ AOB? If so, ask for the minimum value; if not, explain the reason

In the plane rectangular coordinate system, two different moving points a and B on the parabola y = x ^ 2, which are different from the coordinate origin o, satisfy Ao ⊥ Bo (1) Find the trajectory equation of the center of gravity g of △ AOB (2) Is there a minimum value for the area of △ AOB? If so, ask for the minimum value; if not, explain the reason

(1) Let the line AB be y = KX + B. if y = x ^ 2 and y = KX + B, xaxb = - B, yayb = B ^ 2, and because xaxb + yayb = 0, B = 1 or 0, B = 0 should be omitted, so the line AB always passes through the fixed point (0,1). Let the line be y = KX + 1, y = x ^ 2 and y = KX + 1, then XA + XB = k, Ya + Yb = 2, and G be (K / 3,1 / 3). Note that the center of gravity coordinate is one third of the sum of three points, then G is constant on the line y = 1 / 3
(2) S △ AOB = AO * Bo / 2 = under half root [(XA ^ 2 + Ya ^ 2) (XB ^ 2 + Yb ^ 2)]. (XA ^ 2 + Ya ^ 2) (XB ^ 2 + Yb ^ 2) = 2 + XA ^ 2 + XB ^ 2, XA ^ 2 + XB ^ 2 = k ^ 2 + 4, s △ AOB = under half root (4 + K ^ 2). When k = 0, s △ AOB has the minimum value of 1