FInd conditional extrema $z = x+2y$

Question

I need to find conditional extrema of a function.

Here is function:

$z = x + 2y$ when $x^2 + y^2 = 5.$

Answer 1

The gradient of $z$ is $(1,2)$. So the extrema are attained on the intersection between $x^2+y^2=5$ and $y=2x$ giving us $(x, y) = (1,2)$ and $(x, y) = (-1,-2)$. Thus, the extreme values of $z$ are $5$ and $-5$

Answer 2

When the line $x+2y=z$ is tangent to the circle $x^2+y^2=5$, $z$ will attain max/min. $$\frac{|z|}{\sqrt{5}}=\sqrt{s} \implies z_{min.max}=\pm 5.$$