In the plane rectangular coordinate system, the line L and the parabola y ^ 2 = 2x intersect at two points a and B (1) Prove: "if the line L passes through the point t (3,0), then the product of vector OA and vector ob = 3" is a true proposition (2) Write down the inverse proposition of the proposition in (1), judge the truth and explain the reason

In the plane rectangular coordinate system, the line L and the parabola y ^ 2 = 2x intersect at two points a and B (1) Prove: "if the line L passes through the point t (3,0), then the product of vector OA and vector ob = 3" is a true proposition (2) Write down the inverse proposition of the proposition in (1), judge the truth and explain the reason

(1) Let K be nonexistent, ∵ line L pass through t (3,0), ∵ a (3, radical 6) B (3, - radical 6) ∵ vector OA * ob = 3 * 3-6 = 3K exist, let y = kx-3k, substitute y ^ 2 = 2x, and get X1 + x2 = (6K ^ 2 + 2) / K ^ 2, X1 * x2 = 9oa * ob = X1 * x2 + Y1 * y2 = 9 + (kx1-3k) (kx2-3k) = 9 + 6K ^ 2 + 2-3 (6K ^ 2 + 2) + 9K ^ 2 = 3 ∵