如图,已知直线y= x-1与y轴交于点C,将抛物线y=- (x-2) 2 向上平移n个单位(n>0)后与x轴交于A,B两
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如图,已知直线y=![]() ![]() (1)直接写出点C的坐标; (2)当经过C,A,B三点的圆的面积最小时。 ①求n的值; ②在y轴右侧的抛物线上是否存在一点P,使得既与直线y= ![]() |
![]() |
![如图,已知直线y= x-1与y轴交于点C,将抛物线y=- (x-2) 2 向上平移n个单位(n>0)后与x轴交于A,B两](/uploads/image/z/18239649-33-9.jpg?t=%E5%A6%82%E5%9B%BE%EF%BC%8C%E5%B7%B2%E7%9F%A5%E7%9B%B4%E7%BA%BFy%3D+x-1%E4%B8%8Ey%E8%BD%B4%E4%BA%A4%E4%BA%8E%E7%82%B9C%EF%BC%8C%E5%B0%86%E6%8A%9B%E7%89%A9%E7%BA%BFy%3D-+%EF%BC%88x-2%EF%BC%89+2+%E5%90%91%E4%B8%8A%E5%B9%B3%E7%A7%BBn%E4%B8%AA%E5%8D%95%E4%BD%8D%EF%BC%88n%EF%BC%9E0%EF%BC%89%E5%90%8E%E4%B8%8Ex%E8%BD%B4%E4%BA%A4%E4%BA%8EA%EF%BC%8CB%E4%B8%A4)
(1)令x=0,y=0-1=-1,
∴点C的坐标(0,-1);
(2)①平移后二次函数的解析式为y=-
(x-2) 2 +n
由题意知:过C,A,B三点的圆的圆心一定在直线x=2上,点C为定点,
∴当圆的半径等于点C到直线x=2的距离时,圆的半径最小,从而圆的面积最小.此时,圆的半径为2,面积为4π,
设圆心为M,直线x=2与x轴交于点D,连结AM,则AM=2,DM=1,
在Rt△PMD中,AD=
∴点A的坐标是(2-
,0),代入抛物线得n=
,
∴当n=
时,过C,A,B三点的圆的面积最小,最小面积为4π;
(3)如图2,当点P在直线AC下方时,设直线y=
x-1与x轴相交于点E,过点P作PN⊥EC于点N,PM∥y轴交EC于点N,则∠PMN=∠OCE,∠PNM=∠COE=90°,
∴△PMN∽△ECO,
∴
令y=
x-1=0.则x=
,即OE=
,CE=
,
设点P的横坐标为m,则PM=
m-1+
(m-2) 2 -
=
∴PN=
,
根据题意,
=m,解得m 1 =
, m 2 =
(不合题意,舍去)
即点P的坐标是(
)
当点P在直线AC上方时,同理可得
=-m,
解得m 3 =
(不合题意,舍去),
即点P的坐标是(
),
综上,点P的坐标是(
)或(
)。
∴点C的坐标(0,-1);
(2)①平移后二次函数的解析式为y=-
![](http://img.wesiedu.com/upload/e/d8/ed8dd41a228f24cf095f00325a8282c9.jpg)
由题意知:过C,A,B三点的圆的圆心一定在直线x=2上,点C为定点,
∴当圆的半径等于点C到直线x=2的距离时,圆的半径最小,从而圆的面积最小.此时,圆的半径为2,面积为4π,
设圆心为M,直线x=2与x轴交于点D,连结AM,则AM=2,DM=1,
在Rt△PMD中,AD=
![](http://img.wesiedu.com/upload/3/9e/39e00a15f0ae3ef7adbc658101409a13.jpg)
∴点A的坐标是(2-
![](http://img.wesiedu.com/upload/b/61/b611486515abfa0efb297843af6c6cf5.jpg)
![](http://img.wesiedu.com/upload/5/41/54143f6b6c20944addba8fac14c259a2.jpg)
∴当n=
![](http://img.wesiedu.com/upload/8/c5/8c56adcd53781cb6823546bdb1907a9e.jpg)
(3)如图2,当点P在直线AC下方时,设直线y=
![](http://img.wesiedu.com/upload/6/32/6324d941ed92c6a9484acda6e83e525b.jpg)
∴△PMN∽△ECO,
∴
![](http://img.wesiedu.com/upload/0/04/004ac36ed4ede39374907be539d02af0.jpg)
令y=
![](http://img.wesiedu.com/upload/2/cf/2cfe8a922ebf5dc7f2e79f14ce9f4ec3.jpg)
![](http://img.wesiedu.com/upload/6/b6/6b67924b96fca19ad3272da227e32aeb.jpg)
![](http://img.wesiedu.com/upload/1/99/199f33ee2aa499dc6003e92cb7c7399c.jpg)
![](http://img.wesiedu.com/upload/2/af/2af53d4108ad9b9f862e4a5831b78230.jpg)
设点P的横坐标为m,则PM=
![](http://img.wesiedu.com/upload/1/35/135d3f0430044aa925fd68ce46572f07.jpg)
![](http://img.wesiedu.com/upload/e/b4/eb44bf2f634ab1f7bf1f9ff55b8b215d.jpg)
![](http://img.wesiedu.com/upload/b/79/b7909b5590827b72b72040ff854aa4e1.jpg)
![](http://img.wesiedu.com/upload/3/25/32586ada9d645a4a38c912d4973e7e15.jpg)
![](http://img.wesiedu.com/upload/7/01/7011e9f237cfcf7bb3b38df994e7e25f.jpg)
∴PN=
![](http://img.wesiedu.com/upload/c/d1/cd100c70e92a610fb160d1718e624794.jpg)
根据题意,
![](http://img.wesiedu.com/upload/3/4c/34cc5f0d097879dd478a24389c1f0b51.jpg)
![](http://img.wesiedu.com/upload/2/d9/2d916eae46713a81b90003630ac7dd7b.jpg)
![](http://img.wesiedu.com/upload/9/4f/94f133e8c5656bfe0d17c84e82604ecc.jpg)
即点P的坐标是(
![](http://img.wesiedu.com/upload/8/5f/85fbe5ad07ab21647c81fc219cce7b1d.jpg)
当点P在直线AC上方时,同理可得
![](http://img.wesiedu.com/upload/0/82/082be1916b949ba85ca17c9d1bc01f85.jpg)
解得m 3 =
![](http://img.wesiedu.com/upload/2/86/286fc7df8217338fded1c7aba11b97a6.jpg)
![](http://img.wesiedu.com/upload/8/f0/8f0efcffa959b174ed01321097995040.jpg)
即点P的坐标是(
![](http://img.wesiedu.com/upload/5/cf/5cf6f4b009048ee8d1f076950c771070.jpg)
综上,点P的坐标是(
![](http://img.wesiedu.com/upload/6/ce/6cee5cfd6ab75c60ff197ff03ed558fb.jpg)
![](http://img.wesiedu.com/upload/1/d7/1d7455cb25d84a96c33e1da2dce14700.jpg)
![](http://img.wesiedu.com/upload/0/73/07325fa76bdca871ec847e118f233be2.jpg)
![](http://img.wesiedu.com/upload/a/e5/ae525720cc33130b31aa563837fc3acb.jpg)
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