kiwi101 said:Homework Statement
Sketch the line described by the equation:
|z − u| = |z|
z = x+jy
u = −1 + j√3
The Attempt at a Solution
(x+1)^2 + j(y-√3)^2 = (x+jy)^2
I just don't quite get where to go with this
please give me a headstart
kiwi101 said:oh yeah
and um did you mean |z|^2=(x+y)^2?
Dick said:|z|^2=(x^2+y^2). There is no j in there. There shouldn't be any j in the left hand side either. It's an absolute value.
kiwi101 said:oh yeah
and um did you mean |z|^2=(x+y)^2?
kiwi101 said:I was just about to write a long argument about how I was right and then I realized you're right. I misinterpreted something.
So this is what I have done, I feel its right.
(x+1)^2 + (y-√3)^2 = x^2 + y^2
x^2 + 2x + 1 + y^2 -2√3y + 3 = x^2 + y^2
2x + 1 -2√3y + 3 = 0
(x+2)/√3 = y
and then rationalize it and this is the equation of the line?
kiwi101 said:Wait do I solve for y and then rationalize like
y = (-1 -x)/j
kiwi101 said:I just assumed that since it says Im(z) it meant to include the imaginary iota.
So then I guess it is just a line y = -1-x
Complex numbers are numbers that are written in the form a + bi, where a and b are real numbers and i is the imaginary unit equal to the square root of -1.
To sketch the line |z − u| = |z|, first rewrite the equation as |z − u| = |z − 0|. This means that the distance from the complex number z to the complex number u is equal to the distance from z to 0. This creates a circle centered at u with radius equal to the distance from u to 0. The line |z − u| = |z| is the boundary of this circle.
The line |z − u| = |z| represents all the complex numbers that are equidistant from the complex number u and the origin (0,0). It is a circle with center at u and radius equal to the distance from u to 0.
This equation has infinitely many solutions. Any complex number z that is equidistant from u and 0 satisfies the equation. This means that there is a continuum of solutions along the circle, rather than a finite number of solutions.
Yes, the Pythagorean Theorem can be used to solve this equation. Since the equation represents a circle, the distance from z to u and the distance from z to 0 can be represented as the legs of a right triangle, with the distance between u and 0 as the hypotenuse. Using the Pythagorean Theorem, you can find the value of z that satisfies the equation.