Complex Numbers And Quadratic Equations question 357
Question: The region of Argand plane defined by $ |z-1|+|z+1|\le 4 $ is
Options:
A) Interior of an ellipse
B) Exterior of a circle
C) Interior and boundary of an ellipse
D) None of these
Show Answer
Answer:
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.
BETA
