Question

If $z$ is a complex number satisfying 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

Answer: (B)

Given $\left|z+i\right|+\left|z-i\right|=8$ .
$\Rightarrow \left|z-\left(-i\right)\right|+\left|z-i\right|=8$

We know that an ellipse is a locus of a point whose distance from two fixed points is constant and that constant is equal to the length of the major axis.
Here, also locus of $z$ is an ellipse as $zA+zB=8$ (constant)
So, length of the major axis $=8$ .
$\Rightarrow 2a=8\Rightarrow a=4$
Thus, ${\left|z\right|}_{max}=a=4$ .