Complex q (1 Viewer)

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If z1 = 3 + 4i and |z2| = 13, find the greatest value of |z1 + z2|. If |z1 + z2| is at its
greatest value, find the value of z2 in Cartesian form.

I get |z1 + z2|=18 and i get a^2+b^2=169 and (a+3)^2+(b+4)^2=356 but for some reason when i try solving i dont get the answer

 

[Vall]

A graph is definitely useful for this question. From it we can see that the maximum modulus of z1 + z2 will occur when a straight line connects the origin, z1 and z2 (the diagonal orange one). We know that:
theta = arctan(4/3)
using z1. And because the value of z2 giving maximum modulus is also on that line:
theta = arctan(b/a)
tan(theta) = b/a
Then we can use this equation and the one you found (a^2+b^2=169) to solve for a and b.
b^2 = a^2 * tan^2(theta)
Sub in
169 = a^2 * (1+tan^2(theta))
a = +(169 / (1+tan^2(theta)))^0.5 (a > 0 from the graph)
= (169 / (1+tan^2(arctan4/3)))^0.5
= 7.8
Then sub for b
b = +((7.8)^2 * tan^2(arctan(4/3)))^0.5 (b > 0 from the graph)
= 10.4
Quick sanity check: b > a which seems right looking at the graph
Therefore, the value of z2 giving maximum |z1 + z2| is z2 = 7.8 + 10.4i