A) Interior of an ellipse
B) Exterior of a circle
C) Interior and boundary of an ellipse
D) None of these
Correct Answer: C
Solution :
We have \[|z-1|+|z+1|\le 4\] Þ \[|z-1{{|}^{2}}+|z+1{{|}^{2}}+2|z-1||z+1|\,\le 16\] Þ \[(z-1)(\overline{z}-1)+(z+1)(\overline{z}+1)+2|(z-1)(z+1)|\le 16\] Þ \[2|z{{|}^{2}}+2+2|{{z}^{2}}-1|\le 16\] Þ \[|z{{|}^{2}}+|{{z}^{2}}-1|\le 7\] Þ \[|x+iy{{|}^{2}}+|{{(x+iy)}^{2}}-1|\,\,\le 7\]Þ\[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{3}\le 1\] (ellipse) Therefore the points \[z\] are on the boundary or in the interior of the ellipse.
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