【答案】
分析:(1)(2)根據(jù)拋物線的解析式可得出拋物線的對(duì)稱軸和A點(diǎn)坐標(biāo),然后將A點(diǎn)坐標(biāo)代入直線的解析式中進(jìn)行驗(yàn)證即可得出A點(diǎn)是否在直線
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/0.png)
上的.
求∠OAB的度數(shù),可通過求∠OAB的正切值來得出,根據(jù)直線AB的解析式可得出B點(diǎn)坐標(biāo),即可得出OB的長(zhǎng),OA的長(zhǎng)已求出,因此可在三角形OAB中得出∠OAB的正切值.即可得出∠OAB的度數(shù).
(3)本題可分成四種情況:
一:∠AQP=∠AOB=90°:
①AO=PQ,OB=AQ,此時(shí)P、B重合,即可求出P點(diǎn)坐標(biāo)(根據(jù)拋物線的對(duì)稱性可知:P點(diǎn)關(guān)于拋物線對(duì)稱軸的對(duì)稱點(diǎn)也符合要求).
②AO=AQ,PQ=OB,此時(shí)P點(diǎn)縱坐標(biāo)的絕對(duì)值與A點(diǎn)橫坐標(biāo)相等,可將其代入拋物線的解析式中,可得出兩個(gè)符合條件的P點(diǎn)坐標(biāo).
二:∠APQ=∠AOB=90°:
①AO=PA,OB=PQ,可過P作拋物線對(duì)稱軸的垂線,通過∠PAQ的度數(shù)和AP即OA的長(zhǎng)求出P點(diǎn)縱坐標(biāo),然后代入拋物線的解析式中即可得出兩個(gè)符合條件的P點(diǎn)坐標(biāo).
②AO=PQ,PA=OB,同①
因此本題共有8個(gè)符合條件的P點(diǎn)坐標(biāo).
解答:解:(1)對(duì)稱軸:x=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/1.png)
m;
頂點(diǎn):A(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/2.png)
m,0).
(2)將x=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/3.png)
m代入函數(shù)y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/4.png)
x-m,
得y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/5.png)
×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/6.png)
m-m=0
∴點(diǎn)A(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/7.png)
m,0)在直線l上.
當(dāng)x=0時(shí),y=-m,
∴B(0,-m)
tan∠OAB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/8.png)
,
∴∠OAB=30度.
(3)以點(diǎn)P、Q、A為頂點(diǎn)的三角形與△OAB全等共有以下四種情況:
①當(dāng)∠AQP=90°,PQ=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/9.png)
,AQ=m時(shí),
如圖1,此時(shí)點(diǎn)P在y軸上,與點(diǎn)B重合,其坐標(biāo)為(0,-m),
代入拋物線y=-(x-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/10.png)
m)
2
得-m=-3m
2,
∵m>0,
∴m=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/11.png)
這時(shí)有P
1(0,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/12.png)
)
其關(guān)于對(duì)稱軸的對(duì)稱點(diǎn)P
2(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/13.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/14.png)
)也滿足條件.
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/images15.png)
②當(dāng)∠AQP=90°,PQ=m,AQ=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/15.png)
時(shí)
點(diǎn)P坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/16.png)
m-m,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/17.png)
m),
代入拋物線y=-(x-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/18.png)
m)
2
得
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/19.png)
m=m
2,
∵m>0,
∴m=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/20.png)
這時(shí)有P
3(3-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/21.png)
,-3)
還有關(guān)于對(duì)稱軸的對(duì)稱點(diǎn)P
4(3+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/22.png)
,-3).
③當(dāng)∠APQ=90°,AP=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/23.png)
,PQ=m時(shí)
點(diǎn)P坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/24.png)
),代入拋物線y=-(x-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/25.png)
m)
2
得
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/26.png)
m=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/27.png)
m
2,
∵m>0,
∴m=2
這時(shí)有P
5(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/28.png)
,-3)
還有關(guān)于對(duì)稱軸的對(duì)稱點(diǎn)P
6(3
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/29.png)
,-3).
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/images31.png)
④當(dāng)∠APQ=90°,AP=m,PQ=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/30.png)
時(shí)
點(diǎn)P坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/31.png)
),
代入拋物線y=-(x-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/32.png)
m)
2得
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/33.png)
m=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/34.png)
m
2,
∵m>0,
∴m=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/35.png)
這時(shí)有P
7(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/36.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/37.png)
)
還有關(guān)于對(duì)稱軸對(duì)稱的點(diǎn)P
8(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/38.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/39.png)
).
所以當(dāng)m=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/40.png)
時(shí),有點(diǎn)P
1(0,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/41.png)
),P
2(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/42.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/43.png)
);
當(dāng)m=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/44.png)
時(shí),有點(diǎn)P
3(3-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/45.png)
,-3),P
4(3+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/46.png)
,-3);
當(dāng)m=2時(shí),有點(diǎn)P
5(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/47.png)
,-3),P
6(3
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/48.png)
,-3);
當(dāng)m=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/49.png)
時(shí),有點(diǎn)P
7(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/50.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/51.png)
),P
8(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/52.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131021232925256460476/SYS201310212329252564604024_DA/53.png)
).
點(diǎn)評(píng):本題主要考查了二次函數(shù)的相關(guān)知識(shí)以及全等三角形的判定,要注意(3)小題中,要分類討論,將所有的情況都考慮到,以免漏解.