Let-A-B-C-be-three-sets-of-complex-numbers-as-defined-below-A-z-Im-z-1-B-z-z-2-i-3-C-z-Re-1-i-z-2-Let-z-be-any-point-in-A-B-C-and-let-w-be-any-point-satisfy

Question Number 21108 by Tinkutara last updated on 13/Sep/17

Let A, B, C be three sets of complex

numbers as defined below

A = {z : Im z ⩾ 1}

B = {z : ∣ z - 2 - i ∣ = 3}

C = {z : Re ((1 - i) z) = \sqrt{2}} .

Let z be any point in A \cap B \cap C and let

w be any point satisfying ∣ w - 2 - i ∣ <

3 . Then, ∣ z ∣ - ∣ w ∣ + 3 lies between

(1) - 6 and 3

(2) - 3 and 6

(3) - 6 and 6

(4) - 3 and 9

Answered by Tinkutara last updated on 15/Sep/17

G i v e n t h a t ∣ w - 2 - i ∣< 3

\Rightarrow D i s t a n c e b e t w e e n w a n d 2 + i i . e . S

i s s m a l l e r t h a n 3 .

\Rightarrow w i s a p o i n t l y i n g i n s i d e t h e c i r c l e

w i t h c e n t r e S a n d r a d i u s 3 .

\Rightarrow D i s t a n c e b e t w e e n z (i . e . t h e p o i n t P)

a n d w s h o u l d b e s m a l l e r t h a n 6 (t h e

d i a m e t e r o f t h e c i r c l e)

i . e . ∣ z - w ∣< 6

B u t w e k n o w t h a t ∣∣ z ∣ - ∣ w ∣∣<∣ z - w ∣

\Rightarrow∣∣ z ∣ - ∣ w ∣∣< 6 \Rightarrow - 6 <∣ z ∣ - ∣ w ∣< 6

- 3 <∣ z ∣ - ∣ w ∣ + 3 < 9

Commented by Tinkutara last updated on 15/Sep/17