Animated Solution for Mathematics - Complex Numbers: For all complex numbers $z_1,z_2$ satisfying $|z_1|=12$ and $|z_2-3-4i|=5$, the minimum value of $|z_1-z_2|$ is

The problem involves two complex numbers, $z_1$ and $z_2$, defined by the following conditions: $|z_1|=12$ and $|z_2-(3+4i)|=5$.

These equations represent geometric loci on the Argand plane. The first equation, $|z_1|=12$, describes a circle $C_1$ centered at the origin $O(0,0)$ with a radius $R_1=12$.

The second equation, $|z_2-(3+4i)|=5$, describes a circle $C_2$ centered at the point $A(3,4)$ with a radius $R_2=5$.

To understand the relationship between these two circles, we calculate the distance $d$ between their centers $O(0,0)$ and $A(3,4)$:

We now compare the distance $d$ with the radii $R_1$ and $R_2$. We observe that:

Since $d+R_2=10<12$ (where $12$ is $R_1$), the circle $C_2$ lies entirely inside the circle $C_1$.

To find the minimum value of $|z_1-z_2|$, we consider the line passing through the centers of both circles. The minimum distance occurs when $z_1$ and $z_2$ are collinear with the origin and the center $A$.

The distance from the origin to the furthest point on the inner circle $C_2$ along the line connecting the centers is $d+R_2=10$. The distance from the origin to any point $z_1$ on the outer circle $C_1$ is always $R_1=12$.

Therefore, the minimum distance between a point on the outer circle and a point on the inner circle is:

The minimum distance between $z_1$ and $z_2$ is 2.