柱面的母线平行于直线X=Y=Z,准线是曲线{x+y+z=0,x^2+y^2+z^2=1},求该柱面 设准线方程为 x+y-z=0,x-y+z=0, 母线平行于直...
\u67f1\u9762\u7684\u6bcd\u7ebf\u5e73\u884c\u4e8e\u76f4\u7ebfX=Y=Z,\u51c6\u7ebf\u662f\u66f2\u7ebf{x+y+z=0,x^2+y^2+z^2=1},\u6c42\u8be5\u67f1\u9762\u7b80\u5355\u8ba1\u7b97\u4e00\u4e0b\u5373\u53ef\uff0c\u7b54\u6848\u5982\u56fe\u6240\u793a
\u5148\u6c42\u51c6\u7ebf\u65b9\u7a0b\u7684\u65b9\u5411\u5411\u91cf\uff0c\u5373(1,1,-1)*(1,-1,1)=(0,-2,-2)\uff0c\u5199\u6210\u53c2\u6570\u5f0f\u5c31\u662f(0,u,u)
\u67f1\u9762\u65b9\u7a0b\u7684\u53c2\u6570\u5f0f\u5c31\u662f(x,y,z)=(0,u,u)+v(1,1,1)=(v,u+v,u+v)
\u6d88\u53bb\u53c2\u6570\uff0c\u5f97\uff0cy=z
简单计算一下即可,答案如图所示
设M(x1,y1,z1)是准线上一点,而准线是二平面x+y+z=0,x^2+y^2+z^2=1的交线,
故x1+y1+z1=0,(1)
x1^2+y1^2+z1^2=1,(2)
母线方向数为(1,1,1),
经过M点的母线为:(x-x1)/1=(y-y1)/1=(z-z1)/1=t,
则参数方程为:
x1=x-t,
y1=y-t,
z1=z-t,
代入(1)和(2)式,
x-t+y-t+z-t=0,
t=(x+y+z)/3,(4)
(x-t)^2+(y-t)^2+(z-t)^2=1,(5)
由(4)代入(5)式,
(2x-y-z)^2/9+(2y-x-z)^2/9+(2z-x-y)^2/9=1,
∴柱面方程为:(2x-y-z)^2+(2y-x-z)^2+(2z-x-y)^2=9。
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