Question

If Z is any complex number satisfying \[\left| {z - 3 - 2i} \right| \leqslant 2\] , then the minimum value of \[\left| {2z - 6 + 5i} \right|\] is
1. \[2\]
2. \[1\]
3. \[3\]
4. \[5\]

Answer 1

Hint: Firstly try to find a relation between the two given complex numbers. Use the triangle inequality for the given complex number to find a relation for maximum and minimum values as per the requirement. Hence you get the relation. Remember that two complex numbers are equal if and only if both their real and imaginary parts are equal. Hence you get the relation.

Complete step-by-step solution:
Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule $i^2 = −1$ combined with the associative, commutative and distributive laws. Every nonzero complex number has a multiplicative inverse. Complex numbers are naturally thought of as existing on a two-dimensional plane.
We have the complex number \[\left| {2z - 6 + 5i} \right| = 2\left| {z - 3 + \frac{5}{2}i} \right|\]
We can write the given complex number as
\[\left| {z - 3 + \frac{5}{2}i} \right| = \left| {\left( {z - 3 - 2i} \right) + 2i + \frac{5}{2}i} \right|\]
 \[ = \left| {\left( {z - 3 - 2i} \right) + \frac{9}{2}i} \right|\]
Using the triangle inequality we have the following inequality ,
\[\left| {\left( {z - 3 - 2i} \right) + \frac{9}{2}i} \right| \geqslant \left| {\left| {\left( {z - 3 - 2i} \right)} \right| - \left| {\frac{9}{2}i} \right|} \right|\]\[ \geqslant \left| {2 - \frac{9}{2}} \right|\]\[ = \left| {\frac{5}{2}} \right|\]
Therefore we get the inequality \[\left| {z - 3 + \frac{5}{2}i} \right| \geqslant \left| {\frac{5}{2}} \right|\]
Hence we get \[\left| {2z - 6 + 5i} \right| \geqslant 5\]
Hence we get the required inequality.
Therefore option (4) is the correct answer.

Note: We must remember the triangle inequality of complex numbers. Keep in mind that a complex number is a number that can be expressed in the form \[x + iy\] where \[x\] and \[y\] are real numbers and \[i\] is a symbol called the imaginary unit, and satisfying the equation \[{i^2} = - 1\] . Because no "real" number satisfies this equation \[i\] was called an imaginary number. For a complex number \[x + iy\] , \[x\]is called the real part and \[y\] is called the imaginary part.