CircleHard
Question
In the argand plane the inequality |(√3 + i)z - (√2 - i)
|2 + |(√ + i)z + (√3 - i) |2 < 28 represents -
|2 + |(√ + i)z + (√3 - i) |2 < 28 represents -Options
A.The region enclosed by an ellipse of area 8π
B.The region enclosed by circle of radius 4.
C.The interior of a circle of radius √2and centre origin.
D.The region enclosed by an ellipse of area 2π.
Solution
|z|2 = z
|(√3 + i)z - (√ - i)|2 + |(√3 - i) + (√2 + i)||z|2 = z
|(√3 + i)z - (√ - i)|2 + |(√3 - i) + (√2 + i)| + ((√ + i)z + (√3 - i))((√2 - i)) + (√3 + i)z) < 28
⇒ 3z + 4z + 3z + 4z < 28 ⇒ |z|2 < 2
x2 + y2 < 2
|(√3 + i)z - (√ - i)
|2 + |(√3 - i) + (√2 + i)||z|2 = z|(√3 + i)z - (√ - i)
|2 + |(√3 - i) + (√2 + i)| + ((√ + i)z + (√3 - i))((√2 - i)) + (√3 + i)z) < 28⇒ 3z
+ 4z + 3z + 4z < 28 ⇒ |z|2 < 2x2 + y2 < 2
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