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JEE Advanced 2011Paper 1
Exam
JEE Advanced
Year
2011
Paper
Paper 1
Subject
Mathematics
Chapter
Complex Number

JEE Advanced 2011 · Paper 1 · Mathematics · Complex Number

If z is any complex number satisfying |z-3-2 i| ≤ 2 , then the...

If zz is any complex number satisfying ∣z−3−2i∣≤2|z-3-2 i| \leq 2, then the maximum value of ∣2z−6+5i∣|2 z-6+5 i| is

Correct answer

5

Step-by-step solution

Given, ∣z−3−2i∣≤2|z-3-2 i| \leq 2 To find minimum of ∣2z−6+5i∣|2 z-6+5 i| or 2∣z−3+52i∣\quad 2\left|z-3+\frac{5}{2} i\right| Using triangle inequality, i.e. ||z1∣−∣z2∣∣≤∣z1+z2∣z_1|-| z_2|| \leq\left|z_1+z_2\right| ∴∣z−3+52i∣\therefore\left|z-3+\frac{5}{2} i\right| =∣z−3−2i+2i+52i∣=\left|z-3-2 i+2 i+\frac{5}{2} i\right| =∣(z−3−2i)+92i∣≥∣∣z−3−2i∣−92∣=\left|(z-3-2 i)+\frac{9}{2} i\right| \geq|| z-3-2 i\left|-\frac{9}{2}\right| ≥∣2−92∣≥52⇒∣z−3+52i∣≥52\geq\left|2-\frac{9}{2}\right| \geq \frac{5}{2} \Rightarrow\left|z-3+\frac{5}{2} i\right| \geq \frac{5}{2} or ∣2z−6+5i∣≥5\quad|2 z-6+5 i| \geq 5
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