It is known that in the plane rectangular coordinate system xoy, the image of quadratic function y = x2-bx + C (b > 0) passes through point a (- 1, b) and intersects with y axis at point B, and the cotangent value of ∠ ABO is 3. (1) find the coordinates of point B; (2) find the analytic expression of this function; (3) if the top point of this function image is C, prove: ∠ ACB = ∠ abo

It is known that in the plane rectangular coordinate system xoy, the image of quadratic function y = x2-bx + C (b > 0) passes through point a (- 1, b) and intersects with y axis at point B, and the cotangent value of ∠ ABO is 3. (1) find the coordinates of point B; (2) find the analytic expression of this function; (3) if the top point of this function image is C, prove: ∠ ACB = ∠ abo

(1) According to the meaning of the title, we get b = 1 + B + c.. C = - 1.. B (0, - 1); (2) passing through point a as ah ⊥ Y axis, the perpendicular foot is point h.. The cotangent value of ∵ - ABO is 3, ∵ cot ∠ ABO = Bhah = 3. Ah = 1, ∵ BH = 3. ∵ Bo = 1, ∵ ho = 2. ∵ B = 2. ∵ the analytic formula of the function is y = x2-2x-1; (3) from y = x2-2x-1 = (x-1) 2-2, we get the coordinates of vertex C as (1, - 2) ∵ AC = 25, ab = 10, BC＝2，AO＝5，BO=1．∴ACAB＝ABAO＝BCBO＝2．∴△ABC∽△AOB．∴∠ACB=∠ABO．