<s>[INST] How do you find the min/max values of $Arg(z)$ for {$z:|z-(4+3i)|=2$}? How do you find the min/max values of $Arg(z)$ for {$z:|z-(4+3i)|=2$}?
I have drawn a diagram with a circle centre $(4+3i)$ and radius = $2$. I think to find the min/max values of $Arg(z)$ I need to find the points of tangency? I don't understand how to do it only with the diagram and any hints would be much appreciated!
The min possible value of $|z|$ is $3$, and max value of $|z|$ is $7$.
Also, I am on a chapter has more simple questions like this, so I'm looking for a not too complicated way of solving this :)
 [/INST] Let $y=mx$ be the equation of a line tangent to the circle $(x-4)^2 + (y-3)^2 =4$. Then substituting the first into the second, we get
$$
\begin{align}
(x-4)^2 + (mx-3)^2 &= 4 \\
(m^2 + 1) \, x^2 - (6m+8) \, x +21 &= 0
\end{align}
$$
Since the line is tangent to the circle, the discriminant of the above must be zero.
$$
\begin{align}
(6m+8)^2 - 4 (m^2 + 1)(21) &= 0 \\
12m^2 - 24m + 5 &= 0 \\
m &= \frac{6 \pm \sqrt{21}}{6}
\end{align}
$$
Finally, we have $\tan \theta = m$, so
$$\theta_1 = \tan^{-1} \left( \frac{6 - \sqrt{21}}{6} \right) = 0.232$$
and
$$\theta_2 = \tan^{-1} \left( \frac{6 + \sqrt{21}}{6} \right) = 1.055$$
</s>