求曲面z=x²-y²和x² +2y² +3z²=3在点(1,1,0)处的切线及法平面方程
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![求曲面z=x²-y²和x² +2y² +3z²=3在点(1,1,0)处的切线及法平面方程](/uploads/image/z/3760328-56-8.jpg?t=%E6%B1%82%E6%9B%B2%E9%9D%A2z%3Dx%26%23178%3B-y%26%23178%3B%E5%92%8Cx%26%23178%3B+%2B2y%26%23178%3B+%2B3z%26%23178%3B%3D3%E5%9C%A8%E7%82%B9%EF%BC%881%2C1%2C0%EF%BC%89%E5%A4%84%E7%9A%84%E5%88%87%E7%BA%BF%E5%8F%8A%E6%B3%95%E5%B9%B3%E9%9D%A2%E6%96%B9%E7%A8%8B)
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求曲面z=x²-y²和x² +2y² +3z²=3在点(1,1,0)处的切线及法平面方程
求曲面z=x²-y²和x² +2y² +3z²=3在点(1,1,0)处的切线及法平面方程
求曲面z=x²-y²和x² +2y² +3z²=3在点(1,1,0)处的切线及法平面方程
记 F=x^2-y^2-z,G=x^2+2y^2+3z^2-3,
则 F'=2x,F'=-2y,F'=-1; G'=2x,G'=4y,G'=6z,
在点(1,1,0),曲面F,G的法向量分别为 n1={2,-2,-1},n2={1,2,0},
则切线向量是 n1×n2={2,-1,6},
切线方程为 (x-1)2=(y-1)/(-1)=z/6,
法平面方程是 2(x-1)-(y-1)+6z=0,即 2x-y+6z=1.