It is given that $$|z - 1 + i| = 2.$$ What is the maximum possible value of $$|z|?$$ - HSC - SSCE Mathematics Extension 2 - Question 7 - 2024 - Paper 1

This describes a circle centered at the point $(1, -1)$ with a radius of $2$. The equation can be expressed as:

$(x - 1)^2 + (y + 1)^2 = 4.$

To find the maximum value of $|z|$, we need to maximize:

$|z| = ext{distance from the origin to } (x, y),$

which is given by:

$|z| = ext{sqrt}(x^2 + y^2).$

The maximum distance occurs when the circle touches a line passing through the origin.

Using the distance formula and considering the geometry involved, the maximum distance can be found when the line from the origin to $(1, -1)$ is extended. The furthest point on the circle from the origin will be along this line, and we can use the Pythagorean theorem to evaluate the distance:

$|z| = ext{distance} = 2 + ext{radius}.$

Therefore the maximum possible value of $|z| = 2 + ext{distance from origin to circle center} = 2 + ext{sqrt}(1^2 + (-1)^2) = 2 + ext{sqrt}(2) = 2 + ext{sqrt}(2).$$