AP EAMCET · Maths · Complex Number

If $a, b$ are the least and the greatest values respectively of $|z_{1} + z_{2}|,$ where $z_{1} = 12 + 5 i$ and $|z_{2}| = 9,$ then $a^{2} + b^{2} =$

A $468$
B $500$
C $250$
D $450$

Verified Solution

Answer & Solution

Correct Answer

(B) $500$

Step-by-step Solution

Detailed explanation

It is given that z1=12+5i The modulus is Z1=122+52 ⇒Z1=144+25 ⇒Z1=13 The greatest value of z1+z2 a=13+9 ⇒a=22 The lowest value of z1+z2 b=13-9 ⇒b=4

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If a, b are the least and the greatest values respectively of z1+z2, where z1=12+5i and z2=9, then a2+b2=

Step-by-step Solution

If $a, b$ are the least and the greatest values respectively of $|z_{1} + z_{2}|,$ where $z_{1} = 12 + 5 i$ and $|z_{2}| = 9,$ then $a^{2} + b^{2} =$