082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

9.10 Loci and regions of the complex plane 353
and so
thatis
4(x−1) 2 +4y 2 =x 2 +(y−1) 2
3x 2 −8x+3y 2 +2y+3=0
Bycompleting the square thismay be writteninthe form
(
3 x −
3) 4 2 (
+3 y + 1 ) 2
= 8 3 3
thatis
(
x − 4 2 (
+ y +
3) 1 ) 2
= 8 3 9
which represents a circle of radius √ ( 4
8/3 centred at
3 3)
,−1 .
EXERCISES9.10
1 Sketchthe locidefined by
(a) arg(z) = 0
(b) arg(z) = π/2
(c) arg(z −4) = π/4
(d) |2z| = |z −1|
2 Sketchthe regionsdefined by
(a) Re(z) 0
(b) Im(z) < 3
(c) |z| > 3
(d) 0 arg(z) π/2
(e) |z+2|3
(f) |z+j|>3
(g) |z−1|<|z−2|
3 Ifs = σ +jω sketchthe regionsdefined by
(a) σ0
(b)σ0
(c) −2ω2
Solutions
1 SeeFigure S.19.
arg(z) = 0
arg(z) = p/2
arg(z – 4) = p/4
2_
3
– 1_ 1_ 3 3
2|z| = |z – 1|
4
(a)
(b)
(c)
(d)
FigureS.19