Question

If z is a complex number satisfying the equation\[\left| {z + i} \right| + \left| {z - i} \right| = 8\], on the complex plane then maximum value of \[\left| z \right|\] is
A.2
B.4
C.6
D.8

Answer 1

Hint: Complex number is a number generally represented as \[z = a + ib\], where \[a\] and \[b\] is real number represented on real axis whereas \[i\] is an imaginary unit represented on imaginary axis whose value is \[i = \sqrt { - 1} \]. Modulus of a complex number is length of line segment on real and imaginary axis generally denoted by \[\left| z \right|\] whereas angle subtended by line segment on real axis is argument of matrix denoted by argument (z) calculated by trigonometric value. Argument of complex numbers is denoted by \[\arg (z) = \theta = {\tan ^{ - 1}}\dfrac{b}{a}\].

Complete step-by-step answer:
In this question, we need to determine the maximum value of \[\left| z \right|\] such that \[\left| {z + i} \right| + \left| {z - i} \right| = 8\] have to be satisfied. For this we will use the properties of the complex numbers as discussed above.
\[\left| {z + i} \right| + \left| {z - i} \right| = 8 - - (i)\]
This equation can be written as
\[\left| {z - \left( { - i} \right)} \right| + \left| {z - i} \right| = 8 - - (ii)\]
We know imaginary unit \[i\]is represented on a plane as

Now it is given that a point z is on the plane whose sum of distance from points \[i\] and \[ - i\] is given as 8 as shown in the diagram below
 

We know the equation of the locus from point P for the sum of distance between two fixed points is constant \[PA + PB = 2a - - (iii)\]and this constant form an ellipse

Now if we compare equation (iii) with the equation (i), we can say
\[
  2a = 8 \\
  a = 4 \\
 \]
Where \[a = 4\] is the maximum value from which the ellipse pass, hence we can say the maximum value of \[\left| z \right|\] is \[ = 4\]

Note: Complex numbers are always written in the form of \[z = a + ib\] where $a$ and $b$ are real numbers whereas \[i\]being imaginary part. We can convert a degree into radian by multiplying it by \[\dfrac{\pi }{{180}}\]. Argument of complex numbers is denoted by \[\arg (z) = \theta = {\tan ^{ - 1}}\dfrac{b}{a}\].