Find the points on the ellipsoid x22y23z21 where the tangent

Find the points on the ellipsoid x^2+2y^2+3z^2=1 where the tangent plane is parallel to the plane 3x-y+3z=1.

We want (x; y; z) such that x2 +2y2 +3z2 = 1 and < 2x; 4y; 6z >= < 3;-1; 3 >; for some ; that is x = 3=2; y = -=4; z = =2: Thus we must have x2 + 2y2 + 3z2 = (9=4 + 1=8 + 3=4)2 = 1 =) = 2 (x,y,z)=+(3sqrt(5)/5,-sqrt(2)/10,sqrt(2)/2),-(3sqrt(5)/5,-sqrt(2)/10,sqrt(2)/2)