is unimodular complex number and lies on perpendicular bisector of i and −i. ⇒
z
|z|
= ± 1 ⇒ z = ± 1 |z| ⇒ a is real number ⇒Im(z) = 0. (B)→(P) We have |z + 4| + |z - 4| = 10 Where z lies on an ellipse whose focus are (4, 0) and (−4, 0) and length of major axis is 10. Therefore, 2ae = 8 and 2a = 10 ⇒ e =
4
5
Therefore, Re(z) ≤ 5. (C)→(P), (T) We have |w| 2 ⇒ w = 2 (cos θ + i sin θ) x + iy = 2 (cos θ + i sin θ) -
1
2
(cos θ - i sin θ) =
3
2
cosθ+i
5
2
sinθ ⇒
x2
(
3
2
)2
+
y2
(
5
2
)2
= 1 Thus,we have e2 = 1 -
9
4
25
4
= 1 -
9
25
=
16
25
⇒ e =
4
5
(D)→(Q), (T) We have |w| = 1 ⇒ x + iy = cos + i sin θ + cos θ - i sin θ x + iy = 2 cos θ Hence, |Re (z)| ≤ 1, and Im (z) = 0.