complex number

N

nanase

Junior Member
Joined
Aug 8, 2019
Messages
243
complex number.jpg

May I ask some help on how to do part a ?
the answer is abs(z+2i)=2
but I don't get the logic and would really some explanation that leads to that.
TIA
 
nanaseailie said:
View attachment 29652

May I ask some help on how to do part a ?
the answer is abs(z+2i)=2
but I don't get the logic and would really some explanation that leads to that.
TIA
Click to expand...
What I presume is that the equation in a) holds true for every complex number z on the circle. What you have to understand is that |a-b| represents the distance between two complex numbers a and b. Also, a circle with circumference r and center in the point O is the set of all points that are at an r distance from the point O (center of the circle, think of how a compass works!). In this picture, try to find the center of your circle, it will be a complex number (point) at the imaginary axis. Also, try to see what is the circumference of your circle. Then, for every point (complex number) z on the circle, |z-O| = r will hold!
 
Zermelo said:
What I presume is that the equation in a) holds true for every complex number z on the circle. What you have to understand is that |a-b| represents the distance between two complex numbers a and b. Also, a circle with circumference r and center in the point O is the set of all points that are at an r distance from the point O (center of the circle, think of how a compass works!). In this picture, try to find the center of your circle, it will be a complex number (point) at the imaginary axis. Also, try to see what is the circumference of your circle. Then, for every point (complex number) z on the circle, |z-O| = r will hold!
Click to expand...
thank you for the explanation, i get some idea, but why is it |z+2i|=2 as an answer, not |z-2i|=2.
may I also ask why the formula for circle that you wrote is |z-O| = r, not |z+O| = r.
I am still confused with the sign, but I'm getting there!
 
nanaseailie said:
thank you for the explanation, i get some idea, but why is it |z+2i|=2 as an answer, not |z-2i|=2.
may I also ask why the formula for circle that you wrote is |z-O| = r, not |z+O| = r.
I am still confused with the sign, but I'm getting there!
Click to expand...
Just as the distance between two points a and x on the number line (as in your equation of a circle) is |x-a|, the complex-number "distance" (or vector) from O to Z is Z-O, and the actual distance is |Z-O|.

Here, the center, O, is the point (0,-2), or the complex number -2i; and z - (-2i) = z + 2i.
nanaseailie said:
does it have any connection with the circle equation (x-a)^2 + (y-b)^2 = r^2 ?
Click to expand...
Yes, it's the same basic idea, and you could even use this form directly, though working with the complex numbers directly is simpler in some ways. You'd have z = (x,y) and O = (a,b) = (0,-2).
 
Dr.Peterson said:
Just as the distance between two points a and x on the number line (as in your equation of a circle) is |x-a|, the complex-number "distance" (or vector) from O to Z is Z-O, and the actual distance is |Z-O|.

Here, the center, O, is the point (0,-2), or the complex number -2i; and z - (-2i) = z + 2i.

Yes, it's the same basic idea, and you could even use this form directly, though working with the complex numbers directly is simpler in some ways. You'd have z = (x,y) and O = (a,b) = (0,-2).
Click to expand...
My Gosh thank you Dr. Peterson,
your explanation gives some light! I would like to say deeply thank you for your kind time and explanation!
 
You must log in or register to reply here.
Top