在平面直角坐标系中,抛物线过原点O,
\u5728\u5e73\u9762\u76f4\u89d2\u5750\u6807\u7cfb\u4e2d\uff0c\u629b\u7269\u7ebf\u8fc7\u539f\u70b9O\uff0c\u4e14\u4e0ex\u8f74\u4ea4\u4e8e\u53e6\u4e00\u70b9A\uff0c\u5176\u9876\u70b9\u4e3aB\u89e3\uff1a
\uff081\uff09\u629b\u7269\u7ebf\u7684\u5bf9\u79f0\u8f74x=3/2\uff1b
\uff082\uff09\u8bbe\u629b\u7269\u7ebf\u7684\u89e3\u6790\u5f0f\u4e3ay=a(x-3/2)^2+k\uff0c\u4f9d\u9898\u610f\uff0cC\u70b9\u5750\u6807\u4e3a\uff089/2\uff0c9/2+k\uff09\u628aA\u3001C\u4ee3\u5165\uff0c\u89e3\u5f97a=1/2\uff0ck=-9/8\uff0c\u6240\u4ee5\u629b\u7269\u7ebf\u7684\u89e3\u6790\u5f0f\u4e3ay=(1/2)(x-3/2)^2-9/8=x^2/2-3x/2\uff1b
\uff083\uff09\u8fc7E\u4f5cED\u5782\u76f4FG\u4e8eD\u70b9\uff0c\u8bbeH(m,\uff0c0)\uff0cG\uff08n\uff0c0\uff09\u3001\u5219E\uff08m\uff0cm^2/2-3m/2\uff09\u3001F\uff08n\uff0cn^2/2-3n/2\uff09\uff0cn-m=3\uff0c
DF=(n^2/2-3n/2)-(m^2/2-3m/2)=(n-m)(n+m-3)/2=3m
EH+FG=(m^2/2-3m/2)+(n^2/2-3n/2)=(m^2+n^2)/2-3(m+n)/2=m^2
EF^2=DF^2+ED^2=9+9m^2
\u6240\u4ee5(EF^2-9)/6=(9+9m^2-9)/6=3m^2/2
S\u68af\u5f62EFGH=0.5*3*(EH+FG)=0.5*3*m^2=3m^2/2
\u6240\u4ee5S\u68af\u5f62EFGH=(EF^2-9)/6
\u2460\u2235\u629b\u7269\u7ebf\u8fc7\u539f\u70b9O\uff0c\u4e14\u4e0ex\u8f74\u4ea4\u4e8e\u53e6\u4e00\u70b9A\uff08A\u5728O\u53f3\u4fa7\uff09\uff0cOA=3\uff0c\u2234A\u70b9\u5750\u6807\u4e3a\uff083\uff0c0\uff09\uff0c\u2234\u629b\u7269\u7ebf\u7684\u5bf9\u79f0\u8f74\u4e3a\u76f4\u7ebfx=32\uff1b\u2461\u2235\u629b\u7269\u7ebf\u7684\u5bf9\u79f0\u8f74\u4e3a\u76f4\u7ebfx=32\uff0c\u2234\u53ef\u8bbe\u629b\u7269\u7ebf\u7684\u89e3\u6790\u5f0f\u4e3ay=a\uff08x-32\uff092+k\uff0c\u2234\u9876\u70b9B\u7684\u5750\u6807\u4e3a\uff0832\uff0ck\uff09\uff0e\u5982\u56fe1\uff0c\u2235\u70b9C\u7684\u6a2a\u5750\u6807\u4e3a\uff1aON=32+3=92\uff0c\u70b9C\u5728\u629b\u7269\u7ebfy=a\uff08x-32\uff092+k\u4e0a\uff0c\u2234\u70b9C\u7684\u7eb5\u5750\u6807\u4e3aa\uff0892-32\uff092+k=9a+k\uff0e\u2235MC=4.5\uff0c\u22349a+k-k=4.5\uff0c\u2234a=12\uff0c\u5c06A\u70b9\u5750\u6807\uff083\uff0c0\uff09\u4ee3\u5165y=12\uff08x-32\uff092+k\uff0c\u5f9712\uff083-32\uff092+k=0\uff0c\u89e3\u5f97k=-98\uff0c\u2234\u629b\u7269\u7ebf\u7684\u89e3\u6790\u5f0f\u4e3ay=12\uff08x-32\uff092-98\uff0c\u5373y=12x2-32x\uff0e
解：(1)、抛物线的对称轴方程为x=3/2
(2)、由于抛物线过原点，所以可设抛物线的方程为y=ax²+bx
将A(3,0)，代入抛物线方程得：0=9a+3b，即b=-3a————①
当x=3/2时，y=(3/2)²a+(3/2)b
当x=3+(3/2)=9/2时，y=(9/2)²a+(9/2)b
所以：-[(3/2)²a+(3/2)b]+(9/2)²a+(9/2)b=4.5
化简为：12a+2b=3 ————————————————②
解①和②组成的方程组得：a=1/2 , b=-3/2
所以：抛物线的方程为y=(1/2)x²-(3/2)x
(3)、存在。
连接OC，与抛物线对称轴的交点就是D点。此时△BCD的周长就是线段OC和线段AC的长度之和。
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