题目内容
如图,在直角坐标系中,抛物线

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【小题1】直接写出

【小题2】若抛物线与

【小题3】已知点P是直线BC上一个动点,
①当点P在线段BC上运动时(点P不与B、C重合),过点P作PE⊥

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②试探索:在直线BC上是否存在着点P,使得以点P为圆心,半径为

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p;【答案】
【小题1】
.……………………………(2分)
【小题2】由(1)知抛物线为:

∴顶点C坐标为(1,4) ……………………………(3分)
令
∴ B(3,0)……………………(4分)
设直线BC解析式为:
(
),把B、C两点坐标代入,
得
解得
.
∴直线BC解析式为
.……………………(5分)
【小题3】①∵点P(x,y)在
的图象上,

∴PE
,OE
……………………(6分)
∴
PE·OE
∴
………………(7分)
.
符合
,
∴当
时,s取得最大值,最大值为
.……(8分)
②答:存在.
如图,设抛物线的对称轴交x轴于点F,则CF=4,BF=2.

过P作PQ⊥CF于Q,则Rt△CPQ∽Rt△CBF
∴
∴CQ=2r……………(9分)
当⊙P与⊙C外切时,CP
.

解得
舍去).……………(10分)
此时
.……………………(11分)
当⊙P与⊙C内切时,CP
.
.
解得
舍去).……………………(12分)
此时
.
∴当
时,⊙P与⊙C相切.
点P的坐标为
,
.……………………(13分)解析:
略
【小题1】

【小题2】由(1)知抛物线为:

∴顶点C坐标为(1,4) ……………………………(3分)
令

设直线BC解析式为:


得


∴直线BC解析式为

【小题3】①∵点P(x,y)在


∴PE


∴
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
∴
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∴当


②答:存在.
如图,设抛物线的对称轴交x轴于点F,则CF=4,BF=2.

过P作PQ⊥CF于Q,则Rt△CPQ∽Rt△CBF
∴

当⊙P与⊙C外切时,CP


解得

此时

当⊙P与⊙C内切时,CP


解得

此时

∴当

点P的坐标为


略
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