Question

If ${z_1},{z_2},{z_3}$ are 3 distinct complex numbers such that $\dfrac{3}{{\left| {{z_2} - {z_3}} \right|}} = \dfrac{4}{{\left| {{z_3} - {z_1}} \right|}} = \dfrac{5}{{\left| {{z_1} - {z_2}} \right|}}$ then what is the value of $\dfrac{9}{{{z_2} - {z_3}}} + \dfrac{{16}}{{{z_3} - {z_1}}} + \dfrac{{25}}{{{z_1} - {z_2}}}$.
$
  {\text{A}}{\text{. 0}} \\
  {\text{B}}{\text{. }}\sqrt 5 \\
  {\text{C}}{\text{. 5}} \\
  {\text{D}}{\text{. 25}} \\
 $

Answer 1

Hint: Here, we will proceed by finding the values of \[\left( {{z_2} - {z_3}} \right)\], \[\left( {{z_3} - {z_1}} \right)\] and \[\left( {{z_1} - {z_2}} \right)\] from the given equation with the help of formulas like ${\left| z \right|^2} = z\left( {\overline z } \right)$ where z is any complex number and \[\overline {{z_a} - {z_b}} = \overline {{z_a}} - \overline {{z_b}} \] where \[{z_a},{z_b}\] are any two distinct complex numbers.

Complete step-by-step answer:

Given, for any three distinct complex numbers ${z_1},{z_2},{z_3}$
$\dfrac{3}{{\left| {{z_2} - {z_3}} \right|}} = \dfrac{4}{{\left| {{z_3} - {z_1}} \right|}} = \dfrac{5}{{\left| {{z_1} - {z_2}} \right|}} = k{\text{(say) }} \to {\text{(1)}}$
By considering equation (1), we can write
$
  \dfrac{3}{{\left| {{z_2} - {z_3}} \right|}} = k \\
   \Rightarrow k\left| {{z_2} - {z_3}} \right| = 3 \\
 $
By squaring both sides of the above equation, we get
$
   \Rightarrow {\left[ {k\left| {{z_2} - {z_3}} \right|} \right]^2} = {3^2} \\
   \Rightarrow {k^2}{\left| {{z_2} - {z_3}} \right|^2} = 9{\text{ }} \to {\text{(2)}} \\
 $
Again by considering equation (1), we can write
$
  \dfrac{4}{{\left| {{z_3} - {z_1}} \right|}} = k \\
   \Rightarrow k\left| {{z_3} - {z_1}} \right| = 4 \\
 $
By squaring both sides of the above equation, we get
$
   \Rightarrow {\left[ {k\left| {{z_3} - {z_1}} \right|} \right]^2} = {4^2} \\
   \Rightarrow {k^2}{\left| {{z_3} - {z_1}} \right|^2} = 16{\text{ }} \to {\text{(3)}} \\
 $
Again by considering equation (1), we can write
$
  \dfrac{5}{{\left| {{z_1} - {z_2}} \right|}} = k \\
   \Rightarrow k\left| {{z_1} - {z_2}} \right| = 5 \\
 $
By squaring both sides of the above equation, we get
$
   \Rightarrow {\left[ {k\left| {{z_1} - {z_2}} \right|} \right]^2} = {5^2} \\
   \Rightarrow {k^2}{\left| {{z_1} - {z_2}} \right|^2} = 25{\text{ }} \to {\text{(4)}} \\
 $
As we know that for any complex number z, the square of the magnitude of this complex number is equal to the product of this complex number with the conjugate of this complex number.
i.e., ${\left| z \right|^2} = z\left( {\overline z } \right){\text{ }} \to {\text{(5)}}$
Using the formula given by equation (5) in equation (2), we get
\[
   \Rightarrow {k^2}\left( {{z_2} - {z_3}} \right)\left( {\overline {{z_2} - {z_3}} } \right) = 9 \\
   \Rightarrow \left( {{z_2} - {z_3}} \right) = \dfrac{9}{{{k^2}\left( {\overline {{z_2} - {z_3}} } \right)}}{\text{ }} \to {\text{(6)}} \\
 \]
Using the formula given by equation (5) in equation (3), we get
\[
   \Rightarrow {k^2}\left( {{z_3} - {z_1}} \right)\left( {\overline {{z_3} - {z_1}} } \right) = 16 \\
   \Rightarrow \left( {{z_3} - {z_1}} \right) = \dfrac{{16}}{{{k^2}\left( {\overline {{z_3} - {z_1}} } \right)}}{\text{ }} \to {\text{(7)}} \\
 \]
Using the formula given by equation (5) in equation (4), we get
\[
   \Rightarrow {k^2}\left( {{z_1} - {z_2}} \right)\left( {\overline {{z_1} - {z_2}} } \right) = 25 \\
   \Rightarrow \left( {{z_1} - {z_2}} \right) = \dfrac{{25}}{{{k^2}\left( {\overline {{z_1} - {z_2}} } \right)}}{\text{ }} \to {\text{(8)}} \\
 \]
For any two complex numbers \[{z_a},{z_b}\], we can write
\[\overline {{z_a} - {z_b}} = \overline {{z_a}} - \overline {{z_b}} {\text{ }} \to {\text{(9)}}\]
Using the formula given by equation (9) in equation (6), we get
\[ \Rightarrow \left( {{z_2} - {z_3}} \right) = \dfrac{9}{{{k^2}\left( {\overline {{z_2}} - \overline {{z_3}} } \right)}}{\text{ }} \to {\text{(10)}}\]
Using the formula given by equation (9) in equation (7), we get
\[ \Rightarrow \left( {{z_3} - {z_1}} \right) = \dfrac{{16}}{{{k^2}\left( {\overline {{z_3}} - \overline {{z_1}} } \right)}}{\text{ }} \to {\text{(11)}}\]
Using the formula given by equation (9) in equation (8), we get
\[ \Rightarrow \left( {{z_1} - {z_2}} \right) = \dfrac{{25}}{{{k^2}\left( {\overline {{z_1}} - \overline {{z_2}} } \right)}}{\text{ }} \to {\text{(12)}}\]
Let x be the value of the expression which we have to evaluate
i.e., $x = \dfrac{9}{{{z_2} - {z_3}}} + \dfrac{{16}}{{{z_3} - {z_1}}} + \dfrac{{25}}{{{z_1} - {z_2}}}$
By substituting equations (10), (11) and (12) in the above equation, we get
$
   \Rightarrow x = \dfrac{9}{{\left[ {\dfrac{9}{{{k^2}\left( {\overline {{z_2}} - \overline {{z_3}} } \right)}}} \right]}} + \dfrac{{16}}{{\left[ {\dfrac{{16}}{{{k^2}\left( {\overline {{z_3}} - \overline {{z_1}} } \right)}}{\text{ }}} \right]}} + \dfrac{{25}}{{\left[ {\dfrac{{25}}{{{k^2}\left( {\overline {{z_1}} - \overline {{z_2}} } \right)}}} \right]}} \\
   \Rightarrow x = \dfrac{{9{k^2}\left( {\overline {{z_2}} - \overline {{z_3}} } \right)}}{9} + \dfrac{{16{k^2}\left( {\overline {{z_3}} - \overline {{z_1}} } \right)}}{{16}} + \dfrac{{25{k^2}\left( {\overline {{z_1}} - \overline {{z_2}} } \right)}}{{25}} \\
   \Rightarrow x = {k^2}\left( {\overline {{z_2}} - \overline {{z_3}} } \right) + {k^2}\left( {\overline {{z_3}} - \overline {{z_1}} } \right) + {k^2}\left( {\overline {{z_1}} - \overline {{z_2}} } \right) \\
   \Rightarrow x = {k^2}\left[ {\overline {{z_2}} - \overline {{z_3}} + \overline {{z_3}} - \overline {{z_1}} + \overline {{z_1}} - \overline {{z_2}} } \right] \\
   \Rightarrow x = {k^2} \times 0 \\
   \Rightarrow x = 0 \\
 $
Therefore, the value of the expression $\dfrac{9}{{{z_2} - {z_3}}} + \dfrac{{16}}{{{z_3} - {z_1}}} + \dfrac{{25}}{{{z_1} - {z_2}}}$ is 0.
Hence, option A is correct.

Note: Any complex number z can be represented as $z = a + ib$ where a is the real part of the complex number z, b is the imaginary part of the complex number z and $i = \sqrt { - 1} $. All the real numbers are imaginary numbers because any real number r can be represented as $r = r + i\left( 0 \right)$ where the real part is the number itself and the imaginary part is 0.