First slide

Geometry of complex numbers

Question

The region of the Argand diagram defined by $| z - 1 | + | z + 1 | \leq 4$ is

Very difficult

A

Interior of an ellipse
B

Exterior of an ellipse
C

Interior and boundary of an ellipse
D

None of the above

Solution

Let P, A, B represent complex number $z, 1 + 0 i, - 1 + 0 i$ respectively, then
$| z - 1 | + | z - 1 | \leq 4$
$\Rightarrow P A + P B \leq 4$
$\Rightarrow$ P moves in such a way that the sum of its distance from two fixed points is always less than or equal to 4. (Which is greater than the distance between A and B).
$\Rightarrow$ locus of P is the interior and boundary of ellipse having foci at $(1, 0) a n d (- 1, 0) .$

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