JEE Maths›Complex Numbers & Quadratic Equations›Visual Solution
Visual SolutionPYQ 2024Tricky
Question
If ∣z∣=2|z| = 2, the maximum value of ∣z+3+4i∣|z + 3 + 4i| is:
(A)55
(B)77
(C)33
(D)99
Solution Path
By triangle inequality, ∣z+3+4i∣≤∣z∣+∣3+4i∣=2+5=7|z + 3 + 4i| \leq |z| + |3 + 4i| = 2 + 5 = 7.
01Question Setup
1/4
If ∣z∣=2|z| = 2, find the maximum value of ∣z+3+4i∣|z + 3 + 4i|.
max⁡∣z+3+4i∣=  ?\max|z + 3 + 4i| = \;?
02Triangle Inequality
2/4
The triangle inequality states ∣z1+z2∣≤∣z1∣+∣z2∣|z_1 + z_2| \leq |z_1| + |z_2|, with equality when arg⁡(z1)=arg⁡(z2)\arg(z_1) = \arg(z_2).
∣z1+z2∣≤∣z1∣+∣z2∣|z_1 + z_2| \leq |z_1| + |z_2|
03Apply BoundKEY INSIGHT
3/4
∣3+4i∣=9+16=5|3 + 4i| = \sqrt{9 + 16} = 5. So ∣z+3+4i∣≤2+5=7|z + 3 + 4i| \leq 2 + 5 = 7.
Maximum=7\text{Maximum} = 7
04Final Answer
4/4
Maximum value of ∣z+3+4i∣|z + 3 + 4i| is 77.
7\boxed{7}
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