已知过抛物线y^2=4x准线上一点M作两
抛物线y^2=4x①准线为x=-1,
∴设M(-1,m),
切线:y=k(x+1)+m,②
代入①,k^2(x+1)^2+2km(x+1)+m^2=4x,
k^2*x^2+(2k^2+2km-4)x+k^2+2km+m^2=0,③
△/4=(k^2+km-2)^2-k^2*(k^2+2km+m^2)
=k^2*(2km-4-2km-m^2)+(km-2)^2
=-4k^2-4km+4=0,
k^2+km-1=0,④
k1k2=-1,
∴两切线垂直。
由④,③变为k^2*x^2-2x+……=0,
切点坐标:xA=1/k1^2,xB=1/k2^2,
由②,yA=k1(1/k1^2+1)+m...全部
抛物线y^2=4x①准线为x=-1,
∴设M(-1,m),
切线:y=k(x+1)+m,②
代入①,k^2(x+1)^2+2km(x+1)+m^2=4x,
k^2*x^2+(2k^2+2km-4)x+k^2+2km+m^2=0,③
△/4=(k^2+km-2)^2-k^2*(k^2+2km+m^2)
=k^2*(2km-4-2km-m^2)+(km-2)^2
=-4k^2-4km+4=0,
k^2+km-1=0,④
k1k2=-1,
∴两切线垂直。
由④,③变为k^2*x^2-2x+……=0,
切点坐标:xA=1/k1^2,xB=1/k2^2,
由②,yA=k1(1/k1^2+1)+m=(1+k1^2+mk1)/k1,
yB=(1+k2^2+k2m)/k2,
AB的斜率n=[(1+k1^2)/k1-(1+k2^2)/k2]/(1/k1^2-1/k2^2)
=k1k2(k2+k1^2k2-k1-k1k2^2)/(k2^2-k1^2)
=-(1-k1k2)/(k2+k1)
=-2/(k1+k2),
AB的方程为y-(1+k1^2+mk1)/k1=-2/(k1+k2)*[x-1/k1^2],
把焦点坐标(1,0)代入得
-(1+k1^2+mk1)/k1=-2/(k1+k2)*[1-1/k1^2],
去分母,得k1(k1+k2)(1+k1^2+mk1)=2(k1^2-1),
约去k1^2-1,得1+k1^2+k1m=2,
∴k1^2+k1m-1=0,
上式是④,
∴直线AB过焦点。
。收起