[b]证明[/b]      设单叶双曲面的方程为[tex=7.714x2.143]ndVNeInDfYwrsgl2eVPHaRWOK9AJ1tgS0QsTR/bh3EMKl/Y0/Y8KqzhlOnaaJJ4oAWL0xXt2JCtymBzypW75uI5AROm+CkKcX4xN+TsTndk=[/tex][tex=8.286x1.286]5/7LqRpbi6k2PiSszKNsKcQo2UJgn3nwivH7EFoF7SI=[/tex] . 直母线族为（Ⅰ）[tex=9.857x3.643]fnpmC2J6JmQBLyo5NmGAz3jVcwYZMZw0YQ/CFBy2Wa8XL/sGgJwgLX3Q/7umcCdzpXEvFkJD4mGb2cEAJr7sNXP87dx8sYJc7OpPdstfuaMzRxucuD4V30y8su6KqCvM3Xm+Cpm31akMs2xjyorEPA==[/tex]，及（Ⅱ）[tex=10.0x3.929]fnpmC2J6JmQBLyo5NmGAz3jVcwYZMZw0YQ/CFBy2Wa8XL/sGgJwgLX3Q/7umcCdzpXEvFkJD4mGb2cEAJr7sNUXiHRa589dt2btxcF19rAAFScB/Fkev4Ik0nmvv6Uc6c+ApT8lOalvoTSCR8lTRDQ==[/tex]. （1）在（Ⅰ）中任取两条直母线[tex=1.857x1.286]hFuALpaBCR2VqWQRkXxLZA==[/tex]，对应的参数为[tex=6.571x1.286]teFs7RbIjMIlrJvdhGNiEKU9Ipuy1Sr+qbVkqagFaAo=[/tex]，[tex=3.286x1.286]apEOQ1IHGBTfN2YgaFk9vQX+UB/7/lS0ZxkrBpP4pFg=[/tex]，[tex=1.857x1.286]hFuALpaBCR2VqWQRkXxLZA==[/tex]分别经过点[tex=9.643x1.286]p8gc7WQ1wONEFFOOeLjOFjVa8Cc2+wQHLBxboGTRXNZ2OnX/nRJ44HcnyGcpauhCbG3GdgLpkr/znZeQNQhVGw==[/tex]，[tex=9.643x1.286]uj0f7eU0SFBttPl+ws99CYk5CgLaT78pHbbDZC/7eU6QaLAG0v3TWi20Qg2hEwqRfuTkdd10HHj8kPZx4an5oA==[/tex]，方向向 量是[tex=11.0x1.286]qb+1wIjXRJmW10KjL0TmEVtql0Uges08G/jbWU9btjyRdXChPPPm9O/56AN4QjffE6SBbo74yCSHp2Q7XDPeDA==[/tex]，[tex=11.0x1.286]QqVMzhuyNRssdK0J+sHVvnyCfWFjcjw6+p1tqzKDJPQFGFx4jUwUTpQLX4opP3ZRqoe5TbvQfuQz5A/emws64w==[/tex] ，显然两方向不共线，计算混合积[tex=26.571x7.286]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[/tex]所以（Ⅰ）中任意两条直母线[tex=1.857x1.286]hFuALpaBCR2VqWQRkXxLZA==[/tex]异面。同理可得（Ⅱ）中任意两条直母线也异面。（2）在两族直母线中分别任取一条，记为[tex=1.929x1.214]AO6CWj5Zc7lLvaSUqeppjg==[/tex]，对应的参数为[tex=6.0x1.214]ayUKQLgIFRPUSY7kU2bmWwcxlm+9AH91OwOiS+9KMyw=[/tex]，[tex=1.929x1.214]AO6CWj5Zc7lLvaSUqeppjg==[/tex]分别经过点[tex=9.714x1.357]A79hc24ygl3l6FuAP7dnq/IE2vEg526tfsEJ9AKzKKAAvOE6PFeT/mXCYSz5O4Lnu30GuSqtOQFiyvWquKL4Og==[/tex]，[tex=9.714x1.357]4MdQkx70SKbLEuw2W7T1gGMfhi5AzvjdaNCAYvMjlmmWtnfRh6h9BOUuNJb3b0SgRJZZKom15AljyMI2rSWV6g==[/tex]，方向向 量是[tex=11.0x1.286]qb+1wIjXRJmW10KjL0TmEVtql0Uges08G/jbWU9btjyRdXChPPPm9O/56AN4QjffE6SBbo74yCSHp2Q7XDPeDA==[/tex]，v[tex=11.0x1.357]NrciXzmeyUOK6r52T8BgybYvV1i5S5xlGOcvidGiDdkL9+rmRm6xaxz//QSV66VP63vppGUEyAl+R9rRlFhjkw==[/tex]如果[tex=2.643x1.0]w9sRSYC8O4yPd1gUjTYw9Q==[/tex]，由于它们经过同一个点，所以[tex=1.643x1.214]+FfH2Eh4kHr2t34PTpkq/Q==[/tex]共面。如果[tex=3.286x1.286]Y5cgNniJXbw1WiM6u0J5lh+IFJXfMhTVrVQrkztjLkQ=[/tex]，则计算混合积[tex=26.5x5.214]pyTfJal+QJ+Cx6ImRcBW9DN/F5RErP5k9hqi/FIKa1e5RH1pFThktunmeqOHbkLm1PKwkitD6CdUWsOxph4K926rwS7O2QweJnWQY/Y9zTbvLP9zavC/cTrUB4L/vcO+yXZC2mRiJ3gzD2EMsVkwT/k5Oo33xGFbFx+PURq9uvw8Nswl4yp/B0665FSIJBCUrmRmAptbJ1Vr0f4QKO6wB/uzjeKM2ibQF5zyvlFE6XliSRbgL8/6i9NLOVW7bKvVfG3ph7kMTdt8hc7qLo1r0syZwTvMXxlNw6XZmJG+JsO0Dbz6etN3Grvwy0vWi5qiyGOJ7ziCubFjB8dSr0vn0YH6TGLKv/d1ZSSjhA6i4bgwzd0SLff2yHzHCAzu8X0k7oADk1YHIpYI8sss8VeAPjtE1HNofgTjvmbPv1BzandqHwt9R6/WKSqKBoTBvGH1[/tex]所以[tex=1.857x1.286]hFuALpaBCR2VqWQRkXxLZA==[/tex]共面，并且当[tex=5.071x1.286]VYDTXoxF8X8j56cRvxJ452KOGNEO2TIV8sK+s0QRtg8=[/tex]时，[tex=1.857x1.286]hFuALpaBCR2VqWQRkXxLZA==[/tex]平行。