1. The problem statement, all variables and given/known data Consider the hyperboloid of two sheets: z^2=x^2+y^2+1 and a point P(0, 1, 0). Find the shortest distance between the hyperboloid and the point P. Also, find coordinates of all points on the surface for which this distance is attained. 2. Relevant equations 3. The attempt at a solution So, I first used the distance function with the point P. (x-0)^2+(y-1)^2+(z-0)^2 and this simplified (with the use of the equation z^2=x^2+y^2+1 to: 2x^2+2y^2-2y+3=f(x,y) I then took the partial derivatives with respects to x and y fx = 4x fy = 4y-2 These equal 0 when x = 0 and y = 1/2. This is the ONLY critical point. I then found the z value with original function so (0, 1/2, sqrt(5)/2) is the point. I then found the distance to (0,1,0) through subtraction. (0, 1/2, -sqrt(5)/2) = Shortest distance from hyperboloid to point P. Now I'm pretty sure that this is correct, but how do I go about finding the coordinates of all points on the surface for which the distance is attained? I'm struggling to start here. Any tips on this would be great.