题干

如图所示,抛物线y=x2+bx+c与x轴交于A,B两点,与y轴交于点C(0,2),连接AC,若tan∠OAC=2.

(1)求抛物线对应的二次函数的解析式;
(2)在抛物线的对称轴l上是否存在点P,使∠APC=90°?若存在,求出点P的坐标;若不存在,请说明理由;
(3)如图所示,连接BC,M是线段BC上(不与B、C重合)的一个动点,过点M作直线l′∥l,交抛物线于点N,连接CN、BN,设点M的横坐标为t.当t为何值时,△BCN的面积最大?最大面积为多少?

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答案(点此获取答案解析)

(1)∵抛物线y=x2+bx+c过点C(0,2),
∴x=2;
又∵tan∠OAC==2,
∴OA=1,即A(1,0);
又∵点A在抛物线y=x2+bx+2上,
∴0=12+b×1+2,b=-3;
∴抛物线对应的二次函数的解析式为y=x2-

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