Lagrange multiplier - no $g(x,y)$

Question

Using the method of Lagrange multipliers, find the points on the circle $x^2 + y^2 = 4$ that are furthest from and closest to the point $(1, 1)$.

I don't even know where to start, no obvious $g(x,y)$?

Answer 1

Hint:

Consider minimizing the distance of the point $p=(x,y)$ to the point $p_1=(1,1)$ under the constraint that $p\in \{x,y:x^2+y^2=4\}$. This problem writes:

$$ \min_{p}d(p,p_1)\quad \text{subject to: $p\in \{(x,y):x^2+y^2=4\}$} $$

where $d(p,p_1)$ is the euclidean distance between $p$ and $p_1$. Does this help?