Complex Loci

thelostchild · #1 $Old$ September 12th, 2008, 02:23 PM

Hey I've got two questions (no answers so ones more of just double checking some working!)

Firstly can someone have a look over this for me:

Quote:

Find the loci on the complex z-plane which satisfies
$z-c=\rho (\frac{1+it}{1-it})$

where c is complex, $\rho$ is real and t is a parameter satisfying $-\infty < t < \infty$

my reasoning is this

$z-c=\rho (\frac{(1+it)^2}{(1-it)(1+it)})$

$z-c=\rho (\frac{1-t^2}{1+t^2}+\frac{2t}{1+t^2}i)$

if we change the parameter now by letting $t=\tan \frac{\theta}{2}$

we then have

$z-c=\rho (cos \theta + i\sin \theta)$

so this would be a circle of radius $\rho$ centered on c

as $|z-c|=|\rho|$
(does this get around the problem that $\rho$ can be negative?)

or did I screw up somewhere!

and for the second question any chance of a little hint because I cant see where to start with this

Quote:

Find the locus of points in the complex plane which satisfy the following equation
$z=a+bt+ct^2$
where a,b and c are complex and $\frac{b}{c}$ is real

for this ive worked out that the final condition implys that if each of the numbers a,b,c were written as $x_a+y_ai$ etc.
then $x_cy_b=x_by_c$ but I cant see a way to apply it so any chance of a hint

many thanks

Simon

bobak · #2 $Old$ September 13th, 2008, 05:15 AM

Quote:

Originally Posted by thelostchild $View Post$

Hey I've got two questions (no answers so ones more of just double checking some working!)

Firstly can someone have a look over this for me:

my reasoning is this

$z-c=\rho (\frac{(1+it)^2}{(1-it)(1+it)})$

$z-c=\rho (\frac{1-t^2}{1+t^2}+\frac{2t}{1+t^2}i)$

if we change the parameter now by letting $t=\tan \frac{\theta}{2}$

we then have

$z-c=\rho (cos \theta + i\sin \theta)$

so this would be a circle of radius $\rho$ centered on c

as $|z-c|=|\rho|$
(does this get around the problem that $\rho$ can be negative?)

or did I screw up somewhere!

and for the second question any chance of a little hint because I cant see where to start with this

for this ive worked out that the final condition implys that if each of the numbers a,b,c were written as $x_a+y_at$ etc.
then $x_cy_b=x_by_c$ but I cant see a way to apply it so any chance of a hint

many thanks

Simon

Your solution to the first one is fine. however there is a neater approach to the 1+ti business. say $w = 1 + ti$ then $\arg w = \arctan t$ then it is easier to simplify $\frac{w}{w^*}$ . Also I don't see how P being negative would be a problem.

part two should be treated as a vector problem.

firstly $b/c$ is real therefore it may make sense to write $c = \alpha b$ where $\alpha$ is a real constant. so $z = a + bt(1+\alpha t)$ . treating a and b as vectors it should be clear that this is the equation of a line. however pay attention to the range of $t(1+\alpha t)$ .

Bobak