- #1

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## Homework Statement

The cylinder x^2 + y^2 = 1 intersects the plane x + z = 1 in an ellipse. Find the point on the ellipse furthest from the origin.

## Homework Equations

$f(x) = x^2 + y^2 + z^2$

$h(x) = x^2 + y^2 = 1$

$g(x) = x + z = 1$

## The Attempt at a Solution

$\langle 2x, 2y, 2z \rangle = \lambda\langle2x, 2y,0\rangle + \mu\langle1,0,1\rangle$

This results in the equations:

$2x = 2x\lambda + \mu$

$2y = 2y\lambda$

$2z = \mu$

Then $\lambda = 1$, then $2x = 2x + \mu$, then $0 = \mu$, so then $z = 0$.

Then $x + z = 1$, so $x = 1$.

And then $1^2 + y^2 = 1$, so $y = 0$, so then I conclude that the point where this ellipse is furthest from the origin is $(1,0,0)$.

This is wrong. The answer should be $(-1,0,2)$