- #1

RJLiberator

Gold Member

- 1,095

- 63

## Homework Statement

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.

## Homework Equations

## 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.