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共回答了22个问题采纳率:86.4% 举报
(1)a=﹣1,b=﹣2,顶点C的坐标为(﹣1,4);
(2)假设在y轴上存在满足条件的点D,过点C作CE⊥y轴于点E.
由∠CDA=90°得,∠1+∠2=90°.又∠2+∠3=90°,
∴∠3=∠1.又∵∠CED=∠DOA=90°,
∴△CED∽△DOA,∴
![](https://img.yulucn.com/upload/3/49/3490f063d7505b5a6d165e2763271a6c_thumb.jpg)
.
设D(0,c),则
![](https://img.yulucn.com/upload/a/7f/a7ff175ef260c9639511a75b88836eaf_thumb.jpg)
.变形得c
2 ﹣4c+3=0,解之得c
1 =3,c
2 =1.
综合上述:在y轴上存在点D(0,3)或(0,1),
使△ACD是以AC为斜边的直角三角形.
(3)①若点P在对称轴右侧(如图①),
![](https://img.yulucn.com/upload/1/0d/10d9a9afe0f820cc17af46c9d2e43294_thumb.jpg)
只能是△PCQ∽△CAH,得∠QCP=∠CAH.
延长CP交x轴于M,∴AM=CM,∴AM
2 =CM
2 .
设M(m,0),则(m+3)
2 =4
2 +(m+1)
2 ,∴m=2,即M(2,0).
设直线CM的解析式为y=k
1 x+b
1 ,
则
![](https://img.yulucn.com/upload/7/e4/7e4c383add663edceaa019c3454d4085_thumb.jpg)
,解之得
![](https://img.yulucn.com/upload/2/ce/2ce157e0d5ef01179b841e40826af0c7_thumb.jpg)
,
![](https://img.yulucn.com/upload/3/2a/32a89de7d368403777d03154eb7c57fb_thumb.jpg)
.
∴直线CM的解析式
![](https://img.yulucn.com/upload/9/cf/9cffd4678059fab0cb85181a9f9144e9_thumb.jpg)
.
联立
![](https://img.yulucn.com/upload/4/09/4090ee3b2d828fcd536859291ffc5168_thumb.jpg)
,解之得
![](https://img.yulucn.com/upload/c/e6/ce6233849155f6d9019a4b39f30bf033_thumb.jpg)
或
![](https://img.yulucn.com/upload/a/ab/aabc23defd73bbc11fdb84076e324b6f_thumb.jpg)
(舍去).
∴
![](https://img.yulucn.com/upload/d/4e/d4e725696e75f949104f3fe39ae3d9fe_thumb.jpg)
.
②若点P在对称轴左侧(如图②),只能是△PCQ∽△ACH,得∠PCQ=∠ACH.
过A作CA的垂线交PC于点F,作FN⊥x轴于点N.
由△CFA∽△CAH得
![](https://img.yulucn.com/upload/e/6e/e6ea1fbdc9d1baaded584f23955f606b_thumb.jpg)
,
由△FNA∽△AHC得
![](https://img.yulucn.com/upload/6/d5/6d5afdeee2c043fc2f3c553c5f9a609b_thumb.jpg)
.
∴AN=2,FN=1,点F坐标为(﹣5,1).
设直线CF的解析式为y=k
2 x+b
2 ,则
![](https://img.yulucn.com/upload/4/f1/4f100480c9914be2820c52f7e316a243_thumb.jpg)
,
解之得
![](https://img.yulucn.com/upload/a/28/a28224a0ee4a02a02f6c0840308ac683_thumb.jpg)
.
∴直线CF的解析式
![](https://img.yulucn.com/upload/a/f6/af6a84fa89d9bdc8d83d67bc527854bc_thumb.jpg)
.
联立
![](https://img.yulucn.com/upload/5/ed/5ed4b84a319320bb2df346123be94bbd_thumb.jpg)
,解之得
![](https://img.yulucn.com/upload/5/68/568b9c7fc353cdccff1ee9017ff0c738_thumb.jpg)
或
![](https://img.yulucn.com/upload/a/ab/aabc23defd73bbc11fdb84076e324b6f_thumb.jpg)
(舍去).
∴
![](https://img.yulucn.com/upload/a/01/a01a248840c859e38c2d96d45524c945_thumb.jpg)
.
∴满足条件的点P坐标为
![](https://img.yulucn.com/upload/7/a6/7a67cac7955371c924f3e7468360f159_thumb.jpg)
或
![](https://img.yulucn.com/upload/0/a7/0a78624e60d03a3ccdfe7ef2d1b225ea_thumb.jpg)
.
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