Match the following for complex number z with the corresponding locus:
Column I
Column II
I. |z|=1
a. Straight Line
II. |z+2i|+|z−2i|=4
b. Ellipse
III. Re(z2)=4
c. Hyperbola
IV. z+¯¯¯z=4
d. line Segment
e. Circle
A
I−e,II−d,III−c,IV−a
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B
I−b,II−c,III−a,IV−d
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C
I−b,II−d,III−a,IV−c
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D
I−e,II−c,III−a,IV−b
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Solution
The correct option is AI−e,II−d,III−c,IV−a (1) |z|=1⇒|z−0|=1 or x2+y2=1(z=x+iy) circle with center at (0,0) and radius =1 (2) |z+2i|+|z−2i|=4 comparing with |z−z1|+|z−z2|=4 we have z1=−2i,z2=2i also |z1−z2|=|4i|=4 ⇒ summation of distances of z from points 2i and −2i is equal to the distance between 2i and -2i ⇒ z lies on line joining 2iand−2i (3) Re(z2)=Re(x2−y2+2xy)=4
x2−y2=4 which is a hyperbola (4) z+¯¯¯z=4 ⇒2Re(z)=4 ⇒2x=4 x=2