If-Re-z-4-2z-1-1-2-then-z-is-represented-by-a-point-lying-on-1-A-circle-2-An-ellipse-3-A-straight-line-4-No-real-locus-

Question Number 20800 by Tinkutara last updated on 03/Sep/17

If Re (\frac{z + 4}{2 z - 1}) = \frac{1}{2}, then z is represented

by a point lying on

(1) A circle

(2) An ellipse

(3) A straight line

(4) No real locus

Commented by ajfour last updated on 03/Sep/17

R e (z) = \frac{1}{2} .

\Rightarrow s t r a i g h t l i n e .

Answered by ajfour last updated on 03/Sep/17

\Rightarrow \frac{x + 4 + i y}{2 x - 1 + i 2 y} = \frac{1}{2} + i q

\Rightarrow \frac{(x + 4 + i y) (2 x - 1 - i 2 y)}{{(2 x - 1)}^{2} + 4 y^{2}} = \frac{1}{2} + i q

\Rightarrow \frac{(x + 4) (2 x - 1) + 2 y^{2}}{{(2 x - 1)}^{2} + 4 y^{2}} = \frac{1}{2}

o r \frac{2 (x + 4) (2 x - 1) + 4 y^{2}}{{(2 x - 1)}^{2} + 4 y^{2}} = 1

\Rightarrow 2 x - 1 = 0 o r 2 (x + 4) = 2 x - 1

f r o m f i r s t x = R e (z) = \frac{1}{2}

f r o m s e c o n d 2 x + 8 = 2 x - 1

n o s o l u t i o n .

s o R e (z) = \frac{1}{2} I m (z) \neq 0 .

s t r a i g h t l i n e w i t h o n e p o i n t

(z = 1 / 2 + 0 i) r e m o v e d .

Commented by ajfour last updated on 03/Sep/17

d o y o u h a v e a s o l u t i o n ?

Commented by Tinkutara last updated on 03/Sep/17

Thank you very much Sir!

Answered by Tinkutara last updated on 03/Sep/17

Commented by Tinkutara last updated on 03/Sep/17

This is in book .

Commented by ajfour last updated on 03/Sep/17

t h a n k s f o r p o s t i n g .

Commented by Tinkutara last updated on 03/Sep/17