在平面直角坐标系中，点O为坐标原点，抛物线y=ax²-5ax+2与x轴交于A，B两点，与y轴交于点C， 如图，抛物线y=ax2-5ax+4a与x轴相交于点A、B，且...
\uff082014?\u6f6e\u9633\u533a\u6a21\u62df\uff09\u5982\u56fe\uff0c\u629b\u7269\u7ebfy=ax2-5ax+4a\u4e0ex\u8f74\u76f8\u4ea4\u4e8e\u70b9A\u3001B\uff0c\u4e14\u8fc7\u70b9C\uff085\uff0c4\uff09\uff0e\uff081\uff09\u6c42a\u7684\u503c\u548c\u8be5\u629b\uff081\uff09\u628a\u70b9C\uff085\uff0c4\uff09\u4ee3\u5165\u629b\u7269\u7ebfy=ax2-5ax+4a\uff0c\u5f9725a-25a+4a=4\uff0c\u89e3\u5f97a=1\uff0e\u2234\u8be5\u4e8c\u6b21\u51fd\u6570\u7684\u89e3\u6790\u5f0f\u4e3ay=x2-5x+4\uff0e\u2235y=x2-5x+4=\uff08x-52\uff092-94\uff0c\u2234\u9876\u70b9\u5750\u6807\u4e3aP\uff0852\uff0c-94\uff09\uff0e\u7efc\u4e0a\u6240\u8ff0\uff0ca\u7684\u503c\u548c\u8be5\u629b\u7269\u7ebf\u9876\u70b9P\u7684\u5750\u6807\u5206\u522b\u662f1\u3001\uff0852\uff0c-94\uff09\uff1b\uff082\uff09\u2235\u7531\uff081\uff09\u77e5\uff0c\u4e8c\u6b21\u51fd\u6570\u7684\u89e3\u6790\u5f0f\u4e3ay=x2-5x+4=\uff08x-1\uff09\uff08x-4\uff09\uff0e\u5219A\uff081\uff0c0\uff09\uff0cB\uff084\uff0c0\uff09\uff0e\u2234AB=3\uff0e\u53c8C\uff085\uff0c4\uff09\uff0c\u2234\u4e09\u89d2\u5f62ABC\u7684\u9762\u79ef\u662f\uff1a12\u00d73\u00d74=6\uff1b\uff083\uff09\u7531\u629b\u7269\u7ebfy=\uff08x-52\uff092-94\u5148\u5411\u5de6\u5e73\u79fb2\u4e2a\u5355\u4f4d\uff0c\u518d\u5411\u4e0a\u5e73\u79fb3\u4e2a\u5355\u4f4d\uff0c\u5f97\u5230\u7684\u4e8c\u6b21\u51fd\u6570\u89e3\u6790\u5f0f\u4e3ay=\uff08x-52+2\uff092-94+3=\uff08x-12\uff092+34=x2-x+1\uff0c\u5373y=x2-x+1\uff0e
\uff081\uff09\u2235\u629b\u7269\u7ebfy=ax2-5ax+4a\u8fc7\u70b9C\uff085\uff0c4\uff09\uff0c\u22344=25a-25a+4a\uff0c\u89e3\u5f97a=1\uff1b\u2235a=1\uff0c\u2234\u629b\u7269\u7ebf\u7684\u89e3\u6790\u5f0f\u4e3a\uff1ay=x2-5x+4\uff0c\u5373y=\uff08x-52\uff092-94\uff0c\u2234\u9876\u70b9P\u7684\u5750\u6807\u4e3a\uff0852\uff0c-94\uff09\uff1b\uff082\uff09\u2235\u629b\u7269\u7ebf\u7684\u89e3\u6790\u5f0f\u4e3a\uff1ay=x2-5x+4\uff0c\u2234A\uff081\uff0c0\uff09\uff0cB\uff084\uff0c0\uff09\uff0c\u2234AB=4-1=3\uff0c\u2235\u70b9C\uff085\uff0c4\uff09\uff0c\u2234S\u25b3ABC=12\u00d73\u00d74=6\uff0e
⑵抛物线与Y轴交于C(0，2)，直线Y=-1/2X+m过C，得：
2=-1/2×0+m，m=2，
∴直线解析式：Y=-1/X+2，
⑴在直线Y=-1/2X+2中，令Y=0，即-1/2X+2=0，
X=4，∴B(4，0)，
代入抛物线解析式：0=16a-20a+2=0，a=1/2，
∴抛物线解析式：Y=1/2X^2-5/2X+2，
⑶设N(n，1/2n^2-5/2n+2)，
①当∠NBH=∠CBO时，
NH/BH=OC/OB=1/2，
∴|1/2n^2-5/2n+2|/|4-n|=1/2，
|n^2-5n+4|=|4-n|，n^2-5n+4=±(4-n)，
n=0或n=4或n=2或n=4，
其中一点分别为B，舍去，
∴N1(2，-1)，N2(0，2)(与C重合)，
②∠NBH+∠CBO=90°，
BH/NH=1/2，
|4-n|/(1/2n^2-5/2n+2|=1/2，
n^2-5n+4=±(16-4n)，
n=-3或n=5(n=4舍去)，
∴N2(-3，-1)，N3(5，2)。
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