What is the maximum value of |z+3−4i| if |z| is 12?

Question

nick1337 · Accepted Answer

Step 1
Look at the triangle, the vertices of which are the points given in complex notation:
$0 = 0 + 0 i . z = x + y i w = - 3 + 4 i,$
where
$z = x + y i$
lies on the circle
$| z | = 12$
The length of the 3 sides of that triangle are
$| z | = 12 | - 3 + 4 i | = 5 | z - (- 3 + 4 i) | = | z + 3 - 4 i |$
and by the triangle equality we readily get
$| z + 3 - 4 i | \leq | z | + | - 3 + 4 i | = 12 + 5 = 17$
with equality attained for the point z on that circle, lying on the straight line passing through w=-3+4i and 0=0+0i, where the center 0=0+0i of the circle is separating between w and z, that is
$z = \frac{| z |}{| - 3 + 4 i |} (3 - 4 i) = \frac{12}{5} (3 - 4 i) = \frac{36}{5} - \frac{48}{5} i$
In the attached drawing, the blue circle is |z|=12, the green circle is
$| z + 3 - 4 i | = r$
for $7 < r < 17$
and the red circle is
$| z + 3 - 4 i | = 17$
is the largest circle around w=-3+4i which is still meeting the circle |z|=12.

star233 · Accepted Answer

Step 1 Think geometrically. The statement |z| = 12 means that z is located on a circle of radius 12 centered at the origin of the complex plane. Now move the center of the circle to 3-4i. The radius is still 12, and z + 3-4i is a point on this new circle. The expression |z + 3-4i| gives the distance from the origin to the point z + 3-4i, and the maximum such distance is clearly the distance to 3-4i (which is obviously 5) plus the distance from 3-4i to the point on the circle along the same line from the origin as 3-4i, so the maximum distance is 5 + 12 = 17

What is the maximum value of |z+3-4i| if |z| is

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