曲面x2+2y2+z2=1上平行于平面x-y+2z=1的切平面方程为.

Question

曲面x2+2y2+z2=1上平行于平面x-y+2z=1的切平面方程为__________.

Answer 1

设 f(x,y,z) = x^2+2y^2+z^2-1,
则 f 对 x、y、z 的偏导数分别为 f 'x = 2x ,f 'y = 4y ,f 'z = 2z ,
令 (2x,4y,2z) = k(1,-1,2) ,并且 x^2+2y^2+z^2 = 1 ,
可得 (k/2)^2+2(-k/4)^2+k^2=1 ,解得 k = ±2√22/11 ,
所以切点坐标为 ±(√22/11,-√22/22,2√22/11) ,
切平面方程为 (x-√22/11)-(y+√22/22)+2(z-2√22/11) = 0 ,
或 (x+√22/11)-(y-√22/22)+2(z+2√22/11) = 0 .