# Math Help - A point is constrained to a line and a plane. Find closest and farthest points from O

1. ## Lagrange Multiplier help. Find max/min distance from origin.

I have a problem that I need help with. I know I have to use Lagrange multipliers for this question but I am confused about what equation I want to maximize and minimize... Any help is appreciated.

A point is constrained to lie on the surface x2+ y2=1 and the plane 2x+3y+z=1 What are the closest and farthest points from the origin and where do they occur?

I used Lagrange Multipliers for the solution.

I assumed that I wanted to minimize/maximize f(x,y,z)=x2+y2+z2 ... I'm not sure if this is right.
Then I set up the lagrange equation L(x,y,z,lamda1,lamda2)
Then I differentiated wrt all variables and set the equations = 0.
I solved for the values of lamda1 and lamda2 then for x, y and z.

I concluded that the points occur at f(+-2/sqrt(13),+-3/sqrt(13),0)... Originally I thought this was correct because I though of x2+ y2=1 as a circle but I now realize it is a cylinder... Please can someone check this out!

2. ## Re: A point is constrained to a line and a plane. Find closest and farthest points fr

The values for x and y are correct, but note that if x is positive then so is y, and if x is negative then so is y.
Your value for z is wrong though, it doesn't satisfy the second constraint, from which its values, (for each pair of x and y), can be calculated.

3. ## Re: A point is constrained to a line and a plane. Find closest and farthest points fr

Ok... thanks. Also, when solving the equations there was also another point to test where z actually does equal zero. In this case I would just have to solve x^2+y^2=1 and also 2x+3y=1... is this also right?