Discussion Forum

lf z is any complex number satisfying$|z-3-2 i| \leq2$,  then the maximum value of |2z-6+5i| is 

Answer:

Explanation:

Given ,$|z-3-2 i| \leq2$....(i)

 To find  minimum of |2z-6+5i|

 or  $2|z-3+ \frac{5}{2}i|$

 Using triangle inequality

 i.e, $||z_{1}|-|z_{2}|| \leq |z_{1}+z_{2}|$

 $\therefore$    $|z-3+ \frac{5}{2} i|$

 $= |z-3-2i+2i+\frac{5}{2}i|$

 =$|(z-3-2i)+\frac{9}{2} i| \geq |z-3-2i| -\frac{9}{2}|$

 $\geq |2-\frac{9}{2} |\geq \frac{5}{2} \Rightarrow  |z-3+\frac{5}{2}i| \geq \frac{5}{2}$

   or    $|2z-6+5i| \geq 5$