如图,直线l:y=kx+b经过点A(1,0),且与双曲线y=m/x(x>0)交于点B(2,1),过点P(a,a-1)(a>1)作x轴的平行线分别交曲线y=m/x(x>0)和y=-m/x(x<0)于M,N两点.(1)求证:点P在直线y=kx+b上(2)当a为
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![如图,直线l:y=kx+b经过点A(1,0),且与双曲线y=m/x(x>0)交于点B(2,1),过点P(a,a-1)(a>1)作x轴的平行线分别交曲线y=m/x(x>0)和y=-m/x(x<0)于M,N两点.(1)求证:点P在直线y=kx+b上(2)当a为](/uploads/image/z/11297016-0-6.jpg?t=%E5%A6%82%E5%9B%BE%2C%E7%9B%B4%E7%BA%BFl%EF%BC%9Ay%3Dkx%2Bb%E7%BB%8F%E8%BF%87%E7%82%B9A%EF%BC%881%2C0%EF%BC%89%2C%E4%B8%94%E4%B8%8E%E5%8F%8C%E6%9B%B2%E7%BA%BFy%3Dm%2Fx%EF%BC%88x%EF%BC%9E0%EF%BC%89%E4%BA%A4%E4%BA%8E%E7%82%B9B%EF%BC%882%2C1%EF%BC%89%2C%E8%BF%87%E7%82%B9P%EF%BC%88a%2Ca-1%EF%BC%89%EF%BC%88a%3E1%EF%BC%89%E4%BD%9Cx%E8%BD%B4%E7%9A%84%E5%B9%B3%E8%A1%8C%E7%BA%BF%E5%88%86%E5%88%AB%E4%BA%A4%E6%9B%B2%E7%BA%BFy%3Dm%2Fx%28x%26gt%3B0%29%E5%92%8Cy%3D-m%2Fx%28x%26lt%3B0%29%E4%BA%8EM%2CN%E4%B8%A4%E7%82%B9.%EF%BC%881%EF%BC%89%E6%B1%82%E8%AF%81%EF%BC%9A%E7%82%B9P%E5%9C%A8%E7%9B%B4%E7%BA%BFy%3Dkx%2Bb%E4%B8%8A%EF%BC%882%EF%BC%89%E5%BD%93a%E4%B8%BA)
如图,直线l:y=kx+b经过点A(1,0),且与双曲线y=m/x(x>0)交于点B(2,1),过点P(a,a-1)(a>1)作x轴的平行线分别交曲线y=m/x(x>0)和y=-m/x(x<0)于M,N两点.(1)求证:点P在直线y=kx+b上(2)当a为
如图,直线l:y=kx+b经过点A(1,0),且与双曲线y=m/x(x>0)交于点B(2,1),过点P(a,a-1)(a>1)
作x轴的平行线分别交曲线y=m/x(x>0)和y=-m/x(x<0)于M,N两点.
(1)求证:点P在直线y=kx+b上
(2)当a为何值时,△APN是等腰直角形?
(3)是否存在实数a,使得S△AMN=4S△APM?若存在,请求出所有满足条件的a的值;若不存在,请说明理由.
如图,直线l:y=kx+b经过点A(1,0),且与双曲线y=m/x(x>0)交于点B(2,1),过点P(a,a-1)(a>1)作x轴的平行线分别交曲线y=m/x(x>0)和y=-m/x(x<0)于M,N两点.(1)求证:点P在直线y=kx+b上(2)当a为
(1)证明:y=kx+b经过点A(1,0),点B(2,1),有:
0=k+b
1=2k+b
得:k=1,b= -1,故:y=x-1,当x=a时,y=a-1.即点P(a,a-1)(a>1)在直线y=kx+b上.
(2)当a为何值时,△APN是等腰直角形?
y=m/x(x>0)过点B(2,1),则:1=m/2,m=2,
故:y=a-1时,M(2/(a-1),a-1),N(-2/(a-1),a-1),
若AM⊥MN,则2/(a-1)=1,得a=3,a-1=2,
故M(1,2),N(-1,2),知MN=MA,△APN是等腰直角三角形.
若AM=AN,(3-a)²=(a+1)²,a=1,a-1=0无意义(2/(a-1)分母为0无意义)
所以:a=3时,△APN是等腰直角三角形.
(3)是否存在实数a,使得S△AMN=4S△APM?若存在,请求出所有满足条件的a的值;若不存在,请说明理由.
当MN=4PM时,S△AMN=4S△APM,2/(a-1) - [-2/(a-1)] =4(a - 2/(a-1)),a=(1±√13)/2
因a>1故a=(1-√13)/2舍去,所以:a=(1+√13)/2