(14 points) it is known that the vertex of the parabola y = AX2 + BX + C is a (3, - 3), and an intersection point with the X axis is B (1,0). (1) find the analytical formula of the parabola. (2) (2) P is a moving point on the parabola. Find the coordinates of the point P 0 which minimizes the sum of the distances from P to a and B (3) Let the other intersection of the parabola and the x-axis be c. is there a point m on the parabola such that the area of △ MBC is equal to one third of the area of the quadrilateral with points a, P 0, B and C as vertices? If it exists, request the coordinates of all eligible points m; if not, explain the reason

(14 points) it is known that the vertex of the parabola y = AX2 + BX + C is a (3, - 3), and an intersection point with the X axis is B (1,0). (1) find the analytical formula of the parabola. (2) (2) P is a moving point on the parabola. Find the coordinates of the point P 0 which minimizes the sum of the distances from P to a and B (3) Let the other intersection of the parabola and the x-axis be c. is there a point m on the parabola such that the area of △ MBC is equal to one third of the area of the quadrilateral with points a, P 0, B and C as vertices? If it exists, request the coordinates of all eligible points m; if not, explain the reason

Let the analytic formula of parabola be y = a (x-3) ^ 2-3
∵ the intersection of parabola and x-axis is B (1,0)
∴0=a(1-3)^2-3
=>a=3/4
The analytic formula of parabola is: y = (3 / 4) * (x-3) ^ 2-3
That is y = (3x ^ 2) / 4 - (9x) / 2 + 15 / 4

It is known that the vertex coordinates of the parabola y = AX2 + BX + C are (2,1), and the coordinates of the intersection of the parabola and the x-axis are (3,0). Find: (1) the expression of the parabola; (2) the coordinates of the other intersection of the parabola and the X-axis

(1) ∵ the vertex coordinates of the parabola y = AX2 + BX + C are (2,1), ∵ let the analytic formula of the parabola be y = a (X-2) 2 + 1, and (3,0) be substituted into the analytic formula of the function to get: 0 = a (3-2) 2 + 1, and the solution is a = - 1. Therefore, the expression of the parabola is y = - (X-2) 2 + 1; (2) ∵ the vertex coordinates of the parabola y = AX2 + BX + C are (2,1), ∵ its axis of symmetry is a straight line x = 2, ∵ this parabola and X The coordinates of the intersection of the parabola and the x-axis are (3,0) and (1,0)

If the parabola y = x2 + BX + C intersects with the positive half axis of X axis at two points a and B, and intersects with y axis at point C, and the length of line AB is 1 and the area of △ ABC is 1, then the value of B is 1______ ．

The height of AB in ABC is exactly the absolute value of the ordinate of C point, and \\\124; s \\\124; ABC = 12 × 1 124\\\124\\\\\\\124\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\thevalue of B is - 3

It is known that the parabola y = - x + 2 (k-1) x + K + 2 intersects with the X axis at two points a and B, and a is at the positive half axis point of the X axis, B is at the negative half axis of the X axis, Bo = 5ao

Let the equation - x ^ 2 + 2 (k-1) x + K + 2 = 0
The two roots of are R, s | R | 0 s