Find the complex number z satisfying the equations ∣ ∣ z − 12 z − 8 i ∣ ∣ = 5 3 , ∣ ∣ z − 4 z − 8 ∣ ∣ =1 Options: A . &nbsp 6 B . &nbsp 6 ± 8 i C . &nbsp 6+8i, 6+17i D . &nbsp None of these

Answer: Option C : C We have ∣ ∣ z − 12 z − 8 i ∣ ∣ = 5 3 , ∣ ∣ z − 4 z − 8 ∣ ∣ =1 Let z=x+iy, then ∣ ∣ z − 12 z − 8 i ∣ ∣ = 5 3 ⇒ 3|z-12| = 5|z-8i| ⇒ 9 ( x − 12 ) 2 + 9 y 2 = 25 x 2 +25 ( y − 8 ) 2 ⋯ ⋯ (i) ∣ ∣ z − 4 z − 8 ∣ ∣ =1 ⇒ |z-4| = |z-8| ⇒ ( x − 4 ) 2 + y 2 = y 2 + ( x − 8 ) 2 ⇒ x=6 Putting x=6 in (i), we get y 2 -25y+136=0 y=17, 8 Hence z=6+17i or z=6+8i Trick: Check it with options.

Find the complex number z satisfying the equations ∣∣z"

Discussion Forum : Complex Numbers

Question -

Find the complex number z satisfying the equations $∣ ∣ \frac{z - 12}{z - 8 i} ∣ ∣$ = $\frac{5}{3}$ , $∣ ∣ \frac{z - 4}{z - 8} ∣ ∣$ =1

Options:

A . 6

B .

6 \pm 8 i

C . 6+8i, 6+17i

D . None of these

Answer: Option C
:
C

We have $∣ ∣ \frac{z - 12}{z - 8 i} ∣ ∣$ = $\frac{5}{3}$ , $∣ ∣ \frac{z - 4}{z - 8} ∣ ∣$ =1

Let z=x+iy, then

$∣ ∣ \frac{z - 12}{z - 8 i} ∣ ∣$ = $\frac{5}{3}$ $\Rightarrow$ 3|z-12| = 5|z-8i|

$\Rightarrow$ $9 (x - 12)^{2}$ + $9 y^{2}$ = 25 $x^{2}$ +25 $(y - 8)^{2}$ $\dots$ $\dots$ (i)

$∣ ∣ \frac{z - 4}{z - 8} ∣ ∣$ =1 $\Rightarrow$ |z-4| = |z-8|

$\Rightarrow$ $(x - 4)^{2}$ + $y^{2}$ = $y^{2}$ + $(x - 8)^{2}$ $\Rightarrow$ x=6

Putting x=6 in (i), we get $y^{2}$ -25y+136=0

y=17, 8

Hence z=6+17i or z=6+8i

Trick: Check it with options.

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