Question

如圖，直線y=x+2與拋物線y=ax²+bx+6（a≠0）相交于A（，）和B（4，m），點(diǎn)P是線段AB上異于A、B的動點(diǎn)，過點(diǎn)P作PC⊥x軸于點(diǎn)D，交拋物線于點(diǎn)C．                                             

（1）求拋物線的解析式；                                                                       

（2）是否存在這樣的P點(diǎn)，使線段PC的長有最大值？若存在，求出這個最大值；若不存在，請說明理由；                

（3）求△PAC為直角三角形時點(diǎn)P的坐標(biāo)．                                           

                                                              

　                                                                                                      

                                                                                                          

Answer 1

【考點(diǎn)】二次函數(shù)綜合題．                                                                     

【專題】幾何綜合題；壓軸題．                                                              

【分析】（1）已知B（4，m）在直線y=x+2上，可求得m的值，拋物線圖象上的A、B兩點(diǎn)坐標(biāo)，可將其代入拋物線的解析式中，通過聯(lián)立方程組即可求得待定系數(shù)的值．                                         

（2）要弄清PC的長，實(shí)際是直線AB與拋物線函數(shù)值的差．可設(shè)出P點(diǎn)橫坐標(biāo)，根據(jù)直線AB和拋物線的解析式表示出P、C的縱坐標(biāo)，進(jìn)而得到關(guān)于PC與P點(diǎn)橫坐標(biāo)的函數(shù)關(guān)系式，根據(jù)函數(shù)的性質(zhì)即可求出PC的最大值．                    

（3）當(dāng)△PAC為直角三角形時，根據(jù)直角頂點(diǎn)的不同，有三種情形，需要分類討論，分別求解．                  

【解答】解：（1）∵B（4，m）在直線y=x+2上，                                        

∴m=4+2=6，                                                                                     

∴B（4，6），                                                                                   

∵A（，）、B（4，6）在拋物線y=ax²+bx+6上，                                     

∴，解得，                                              

∴拋物線的解析式為y=2x²﹣8x+6．                                                        

                                                                                                          

（2）設(shè)動點(diǎn)P的坐標(biāo)為（n，n+2），則C點(diǎn)的坐標(biāo)為（n，2n²﹣8n+6），                  

∴PC=（n+2）﹣（2n²﹣8n+6），                                                           

=﹣2n²+9n﹣4，                                                                                 

=﹣2（n﹣）²+，                                                                        

∵PC＞0，                                                                                         

∴當(dāng)n=時，線段PC最大且為．                                                       

                                                                                                          

（3）∵△PAC為直角三角形，                                                                

i）若點(diǎn)P為直角頂點(diǎn)，則∠APC=90°．                                                   

由題意易知，PC∥y軸，∠APC=45°，因此這種情形不存在；                        

ii）若點(diǎn)A為直角頂點(diǎn)，則∠PAC=90°．                                                  

如答圖3﹣1，過點(diǎn)A（，）作AN⊥x軸于點(diǎn)N，則ON=，AN=．                

過點(diǎn)A作AM⊥直線AB，交x軸于點(diǎn)M，則由題意易知，△AMN為等腰直角三角形，              

∴MN=AN=，∴OM=ON+MN=+=3，                                                

∴M（3，0）．                                                                                  

設(shè)直線AM的解析式為：y=kx+b，                                                          

則：，解得，                                                             

∴直線AM的解析式為：y=﹣x+3  ①                                                     

又拋物線的解析式為：y=2x²﹣8x+6 ②                                                    

聯(lián)立①②式，解得：x=3或x=（與點(diǎn)A重合，舍去）                                   

∴C（3，0），即點(diǎn)C、M點(diǎn)重合．                                                         

當(dāng)x=3時，y=x+2=5，                                                                        

∴P₁（3，5）；                                                                                 

              

iii）若點(diǎn)C為直角頂點(diǎn)，則∠ACP=90°．                                                  

∵y=2x²﹣8x+6=2（x﹣2）²﹣2，                                                             

∴拋物線的對稱軸為直線x=2．                                                               

如答圖3﹣2，作點(diǎn)A（，）關(guān)于對稱軸x=2的對稱點(diǎn)C，                          

則點(diǎn)C在拋物線上，且C（，）．                                                      

當(dāng)x=時，y=x+2=．                                                                          

∴P₂（，）．                                                                              

∵點(diǎn)P₁（3，5）、P₂（，）均在線段AB上，                                          

∴綜上所述，△PAC為直角三角形時，點(diǎn)P的坐標(biāo)為（3，5）或（，）．                    

【點(diǎn)評】此題主要考查了二次函數(shù)解析式的確定、二次函數(shù)最值的應(yīng)用以及直角三角形的判定、函數(shù)圖象交點(diǎn)坐標(biāo)的求法等知識．