已知點A（m、n）是反比例函數（x＞0）的圖象上一點，過A作AB⊥x軸於點B，P是y軸上一點，        ...

問題詳情：

已知點A（m、n）是反比例函數（x＞0）的圖象上一點，過A作AB⊥x軸於點B，P是y軸上一點，                                                       

（1）求△PAB的面積；                                                                      

（2）當△PAB為等腰直角三角形時，求點A的座標；                                     

（3）若∠APB=90°，求m的取值範圍．                                              

【回答】

解：（1）連接OA，                                                                          

∵AB⊥x軸，                                                                                     

∴AB∥y軸，                                                                                     

∴S△PAB=S△POB，                                                                              

∵點A（m、n）是反比例函數（x＞0）的圖象上一點，                            

∴S△PAB=S△POB=2；                                                                          

（2）若∠ABP=90°，則AB=OB，                                                        

則m=n，                                                                                           

∴m=，                                                                                           

∵x＞0，                                                                                            

∴m=2，                                                                                            

∴點A（2，2）；                                                                              

若∠PAB=90°，則PA=AB，同理可得點A（2，2）；                                       

若∠APB=90°，則AP=BP，                                                                 

過點P作PC⊥AB於點C，則AC=BC=PC，                                         

則點A（m，2m），                                                                          

∴2m=，                                                                                          

∵x＞0，                                                                                            

∴m=，                                                                                         

∴點A（，2）；                                                                       

綜上，點A的座標為：（2，2）或（，2）；                                         

（3）∵∠APB=90°，                                                                         

∴點P是以AB為直徑的圓與y軸的交點，                                          

由（2）可知當x=時，以AB為直徑的圓與y軸相切，當x＞時，以AB為直徑的圓與y軸相離，                                                           

∴m的取值範圍為：0＜m≤．                                                         

知識點：反比例函數

題型：解答題