The parabola passes through the origin o, the point a (6,8), and the point (3, - 5). If the line BC parallel to the Y-axis of point B on the line OA intersects with the parabola at C, △ OBC isosceles Find the coordinates of point C | Math enthusiast | Mathematical art

The parabola passes through the origin o, the point a (6,8), and the point (3, - 5). If the line BC parallel to the Y-axis of point B on the line OA intersects with the parabola at C, △ OBC isosceles Find the coordinates of point C

Let the analytic formula of parabola be y = ax ^ 2 + BX + C, substituting the points (0,0), (6,8), (3, - 5) into the analytic formula, C = 0 36a + 6B = 8 9A + 3B = - 5, solving the equations, a = 1, B = - 14 / 3, C = 0, the analytic formula of parabola is y = x ^ 2-14 / 3x, the analytic formula of straight line OA is y = 4 / 3x, and point B (x, 4 / 3x) is set

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1. If the line L passing through the point (0, - 1 / 2) intersects with the parabola: y = - x * x at two points a and B, and O is the origin of the coordinate, then the value of the vector of OA multiplied by OB is?
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8. As shown in the figure, y = - 5x + 5 intersects with the coordinate axis at two points a and B, △ ABC is an isosceles right triangle, and the hyperbola y = KX (x < 0) passes through point C to find the value of K
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10. As shown in the figure, if the line y = KX + B (K ≠ 0) intersects the X axis at point a (5 / 2,0), and the hyperbola y = m / X (m ≠ 0) intersects at point B in the second quadrant, and OA = ob The area of △ OAB is 5 / 2 1. Find the analytic formula of straight line AB and hyperbola, 2. If the other intersection of the straight line AB and the hyperbola is the point D, find the value of s △ BOC
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14. Points a and B are two moving points beyond the origin of the parabola y ^ 2 = 2px (P > 0), and OA is perpendicular to ob, Let a (x1, Y1), B (X2, Y2) find the values of Y1 * Y2 and X1 * x2
15. It is known that the line and parabola y2 = 2px (P > 0) intersect at two points a and B, and OA ⊥ ob, OD ⊥ AB intersect at D, and the coordinate of point D is (2,1), then the value of P is () A. 52B. 23C. 54D. 32
16. As shown in the figure, it is known that a straight line and a parabola y ^ 2 = 2px intersect with two points a and B, and OA is perpendicular to ob, OD is perpendicular to AB, and ab intersects with point D. the coordinates of point D are (2,1), and the value of P is obtained
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18. Given that the straight line and parabola y ＾ 2 = 2px intersect at two points AB, and OA ⊥ ob, OD ⊥ AB intersect at D, D coordinate is (2,1), find the value of P First, I calculate the slope k of OD with the coordinates of point D, and then calculate the slope k 'of ab. because AB passes through D, we can know the equation of ab Then, the equation AB is substituted into the parabola, and the x 1 + x 2 and X 1x 2 are calculated by the Weida theorem Then, let a (x1, root 2px1), B (X2, root 2px2), calculate the algebraic formula of OA, OB length, because OA ⊥ ob, so the area is oaob / 2 Then calculate the length of AB section parabola, OD length and area abod / 2 with chord length formula Finally, because the two areas are equal, abod / 2 = oaob / 2, and then combined with vedadine to understand P
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