A straight line with a slope of 1 intersects with a parabola y2 = 2x at two different points a and B. find the trajectory equation of the midpoint m of the line ab

Let the coordinate of M be (x, y), the linear equation of slope 1 be y = x + m, and a (x1, Y1), B (X2, Y2), y be eliminated by y = x + my2 = 2x, then x2 + (2m-2) x + M2 = 0 (2) according to the relationship between the root and the coefficient of the quadratic equation of one variable, we get that X1 + x2 = 2 − 2mx & nbsp; 1x2 = m2 (6 points) ∵ point m is the middle point of line AB, ∵ x = X1 + X22 = 1 − m, y = x + M = 1 (8 points) ∵ there are two different intersection points between straight line and parabola, ∵ △ = (2m-2) 2-4m2 ＞ 0, the solution is m ＜ 12, combined with x = 1-m, the abscissa range of M is (12, + ∞) Therefore, the trajectory equation of point m in line AB is y = 1 (x ∈ (12, + ∞)) (10 points)

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