Question

The minimum value of $\left| {Z - 1 + 2i} \right| + \left| {4i - 3 - Z} \right|$ is
a.$\sqrt 5 $
b.5
c.$2\sqrt {13} $
d.$\sqrt {15} $

Answer 1

Hint: At first let's rearrange the given expression and we can see that it is a sum of the distances of Z from two points P and Q and the sum is minimum in the line segment PQ and it is given by \[PQ = \sqrt {{{\left( {{\text{ real part of P - Q}}} \right)}^2} + {{\left( {{\text{imaginary part of P - Q}}} \right)}^2}} \]

Complete step-by-step answer:
Lets rearrange the given expression as $\left| {Z - \left( {1 - 2i} \right)} \right| + \left| {Z - \left( { - 3 + 4i} \right)} \right|$
From this we can see that the given expression is the sum of the distances of Z from two points
Here the two points are $P = 1 - 2i$ and $Q = - 3 + 4i$
The sum is minimum when Z lies on the line segment PQ
Now to find PQ
\[ \Rightarrow PQ = \sqrt {{{\left( {{\text{ real part of P - Q}}} \right)}^2} + {{\left( {{\text{imaginary part of P - Q}}} \right)}^2}} \]
$
   \Rightarrow PQ = \sqrt {{{\left( {1 + 3} \right)}^2} + {{\left( { - 2 - 4} \right)}^2}} \\
   \Rightarrow PQ = \sqrt {{{\left( 4 \right)}^2} + {{\left( { - 6} \right)}^2}} \\
   \Rightarrow PQ = \sqrt {16 + 36} \\
   \Rightarrow PQ = \sqrt {52} = \sqrt {4*13} = 2\sqrt {13} \\
$
Hence we get the minimum to be $2\sqrt {13} $
Therefore the correct option is c.

Note: The square root of i has both real and imaginary parts. The square root of a negative real number is purely imaginary, but the square root of a purely imaginary number has to have both real and imaginary parts
Any root of i has multiple unique solutions, and the N-th root has N unique solutions. For positive, real numbers, taking the square root (i.e., the second root) of that number gives you two possible solutions: a positive one and a negative one.
In an imaginary fraction, it actually matters whether the numerator or denominator has the “i” in it.