Lagrange multiplier optimizing a 3-D ellipse with respect to the origin

Question

I cannot solve this question:

The plane $x+y+2z=2$ intersects the paraboloid $z=x^2+y^2$ in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin.

My working is as follows:

A different method is attempted, but abandoned:

(Exercise 11.8.41 from Stewart, J. Calculus: concepts and contexts. 4th ed.)

Answer 1

You made some mistakes in method 1.

1. Equation (6) should be $2x+2y=2\lambda+\mu(2x+2y)$

2. Equation (7) has a similar mistake.

3. I didn't understand what you really did when you do "(6) in (7)".

Method 1 seems not work. Try this:

Transform equation (1),(2),(3): $$x=\frac{\lambda}{2(1-\mu)}\\ y=\frac{\lambda}{2(1-\mu)}\\ z=\frac{2\lambda-\mu}{2}$$

You should argue that $\mu\ne 1$ before you proceed. So this gives you $x=y$. Use this information in equation (4),(5), you should be able to find them.

Answer 2

I used KittyL's suggestion to isolate x and y: