Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The plane $x + y + 2z = 30$ intersects the paraboloid $z = x^2 + y^2$ in an ellipse.

Find the points on the ellipse that are nearest to and farthest from the origin.

I know you have to find the gradient for $f$, but after that, I have no clue how to proceed. Thank you in advance.

share|improve this question

1 Answer 1

let $d^2(x,y,z)=x^2+y^2+z^2$.

you want $\nabla (d^2)$ to be in the plane of $\nabla f$ and $\nabla g$. That is, for some $\lambda$ and $\mu$, you want $$\nabla(d^2)+\lambda\nabla f+\mu\nabla g=0.$$

share|improve this answer
    
So you get:<2x, 2y, 2z> + λ<1,1,2> + μ<2x, 2y, -1> = 0 –  Jason Mar 28 '13 at 3:04
    
Now what do I do? –  Jason Mar 28 '13 at 3:06
    
Let's do a confidence check, that is, let's count unknowns, and pray that we have as many equations. Our unknowns are $x,y,z,\lambda,\mu$, five unknowns. We have the three equations you write in the comment above, plus $f=g=0$, two more, so that's five equations, so there's hope. Let's smash them together and pray. –  Brady Trainor Mar 28 '13 at 3:09
    
a thought, four of the equations are linear, might apply some linear algebra to those, I would think that would organize the work some. if my intuition is wrong, let us know what goes wrong please. –  Brady Trainor Mar 28 '13 at 3:13
1  
oops, they're not linear, because the way $\lambda$ and $\mu$ show up. –  Brady Trainor Mar 28 '13 at 3:15

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.