Question

The region of the argand diagram defined by $\left|z-1\right|+\left|z+1\right|$ less than or equal $4$ is an

• A. interior of an ellipse
• B. exterior of a circle
• C. interior and boundary of an ellipse
• D. None of the above

Answer 1

The correct option is C

interior and boundary of an ellipse

Explanation for the correct answer:

$\left|z-1\right|+\left|z+1\right|\le 4$ is an equation of the type.

$\left|z-a\right|+\left|z+a\right|\le 4a$ $...\left(i\right)$

Equation $\left(i\right)$ represents the locus of $z$ such that the sum of its distance from 2 fixed points is less than or equal to a constant value.

By definition of an an ellipse $PS+PS\text{'}=2a$ where, $P$ is any point on the ellipse and $S,S\text{'}$ are the foci.

Thus $\left|z-1\right|+\left|z+1\right|=4$ is an ellipse.

Thus the region $\left|z-1\right|+\left|z+1\right|\le 4$ represents the interior and the boundary of an ellipse.

Hence, option $\left(C\right)$ is the correct answer.