Match the following for complex number $z$ with the corresponding locus:

Column I Column II
I. $| z | = 1$
a. Straight Line
II. $| z + 2 i | + | z - 2 i | = 4$ b. Ellipse
III. $R e (z^{2}) = 4$
c. Hyperbola
IV. $z + ¯ ¯ ¯ z = 4$ d. line Segment
e. Circle

I - e, I I - d, I I I - c, I V - a

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I - b, I I - c, I I I - a, I V - d

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I - b, I I - d, I I I - a, I V - c

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I - e, I I - c, I I I - a, I V - b

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Solution

The correct option is A $I - e, I I - d, I I I - c, I V - a$
(1) $| z | = 1 \Rightarrow | z - 0 | = 1$ or $x^{2} + y^{2} = 1 (z = x + i y)$
circle with center at (0,0) and radius =1
(2) $| z + 2 i | + | z - 2 i | = 4$
comparing with
$| z - z_{1} | + | z - z_{2} | = 4$
we have
$z_{1} = - 2 i, z_{2} = 2 i$
also $| z_{1} - z_{2} | = | 4 i | = 4$
$\Rightarrow$ summation of distances of z from points
$2 i$ and $- 2 i$ is equal to the distance
between 2i and -2i
$\Rightarrow$ z lies on line joining $2 i a n d - 2 i$
(3) $R e (z^{2}) = R e (x^{2} - y^{2} + 2 x y) = 4$
$x^{2} - y^{2} = 4$ which is a hyperbola
(4) $z + ¯ ¯ ¯ z = 4$
$\Rightarrow 2 R e (z) = 4$
$\Rightarrow 2 x = 4$
$x = 2$

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Column I	Column II
I. $\| z \| = 1$	a. Straight Line
II. $\| z + 2 i \| + \| z - 2 i \| = 4$	b. Ellipse
III. $R e (z^{2}) = 4$	c. Hyperbola
IV. $z + ¯ ¯ ¯ z = 4$	d. line Segment
	e. Circle

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

Match the following for complex number $z$ with the corresponding locus:

Column I Column II
I. $| z | = 1$
a. Straight Line
II. $| z + 2 i | + | z - 2 i | = 4$ b. Ellipse
III. $R e (z^{2}) = 4$
c. Hyperbola
IV. $z + ¯ ¯ ¯ z = 4$ d. line Segment
e. Circle