設(shè)二次函數(shù)y=-x2+(m-2)x+3(m+1)的圖象與x軸交于A、B兩點(diǎn)(A在B左邊),與y軸交于C點(diǎn),線段AO與OB的長(zhǎng)的積等于6(O是坐標(biāo)原點(diǎn)),連接AC、BC,求sinC的值.
【答案】
分析:設(shè)二次函數(shù)y=-x
2+(m-2)x+3(m+1)的圖象與x軸的兩交點(diǎn)A、B的坐標(biāo)為(x
1,0),(x
2,0),那么OA=|x
1|,OB=|x
2|,根據(jù)根與系數(shù)的關(guān)系可以得到x
1+x
2=-(m-2),x
1x
2=-3(m+1),而線段AO與OB的長(zhǎng)的積等于6,由此可以求出m,也就求出了拋物線的解析式,然后求出A、B、C三點(diǎn)坐標(biāo),最后利用三角函數(shù)的定義即可求出sinC的值.
解答:
解:∵二次函數(shù)y=-x
2+(m-2)x+3(m+1)的圖象與x軸交于A、B兩點(diǎn)(A在B左邊),
∴設(shè)A、B的坐標(biāo)為(x
1,0),(x
2,0),
∴OA=|x
1|,OB=|x
2|,
∴x
1+x
2=-(m-2),x
1x
2=-3(m+1),
而線段AO與OB的長(zhǎng)的積等于6,
∴3(m+1)=±6,
∴m=1或-3,
當(dāng)m=1時(shí),拋物線解析式為y=-x
2-x+6,
∴A、B的坐標(biāo)為(-3,0),(2,0),C(0,6)
∴AC=3

,BC=2

,AB=5,

如圖拋物線過(guò)A作AD⊥BC于D,
則S
△ABC=

CO•BA=

AD•BC,
∴AD=

=

,
∴sinC=

=

;
當(dāng)m=-3時(shí),拋物線解析式為y=-x
2-5x-6,
∴A、B的坐標(biāo)為(-3,0),(-2,0),C(0,-6)
∴AC=3

,BC=2

,AB=1,
如圖拋物線過(guò)A作AD⊥BC于D,
則S
△ABC=

CO•BA=

AD•BC,
∴AD=

=

,
∴sinC=

=

;
所以sinC的值為

或

.
點(diǎn)評(píng):此題主要考查了拋物線與x軸的交點(diǎn),也考查了三角函數(shù)的定義、根與系數(shù)的關(guān)系和待定系數(shù)法確定函數(shù)的解析式,綜合性比較強(qiáng),對(duì)學(xué)生的能力要求比較高,平時(shí)應(yīng)該加強(qiáng)訓(xùn)練.