1. The problem statement, all variables and given/known data A curve C in space is defined implicitly on the cylinder x^2+y^2=1 by the additional equation: x^2-xy+y^2-z^2=1. Find the point or points on C closest to the origin. 2. Relevant equations d = ((x-x0)+(y-y0)+(z-z0))^(1/2) - This is the distance formula. Please note that I did NOT learn Lagrange Multipliers, yet - it is the next section in my math book. 3. The attempt at a solution First, I used the distance formula: ((x-x0)^2+(y-y0)^2+(z-z0)^2)^(1/2). where (x0,y0,z0) = (0,0,0) - the origin. I solved for z from the additional equation: x^2-xy+y^2-z^2=1. So the equation now looks like this: ((x^2+y^2+(x^2-xy+y^2-1))^(1/2) I then square the entire equation (so that I can derive easier) because the extrema points remained the same if the equation were not squared. I solved the partials for x and y. And for some reason, I got fx=4x-y and fy=4y-x. Thus the point of intersection, which I believe to be incorrect, is (0,0). I plugged (0,0) back into the additional equation and I have square root by a negative number! And now... I am stuck. Can anyone please show me how to approach this problem using the derivation of the distance formula and NOT using Lagrange Multipliers?