Complex loci (2 Viewers)

[QZP]

Hi, can someone help me with the following questions

Illustrate on the Argand diagram the following regions:
a) 1 <= |z| <= |z-i|
b) |z+i| + |z-i| = 4

for a) I had two circles of radius 1 with centres 0,0 and 0,1 however I don't know how to go from here...

 

[bottleofyarn]

Hi, can someone help me with the following questions

Illustrate on the Argand diagram the following regions:
a) 1 <= |z| <= |z-i|
b) |z+i| + |z-i| = 4

for a) I had two circles of radius 1 with centres 0,0 and 0,1 however I don't know how to go from here...

It might help if you split up the inequality and use the z = x + iy form, then find the intersection of the two.
gives you everything outside the circle with centre the origin.
gives you a straight line from


which is everything below a horizontal line, and you can combine the two.

 

[braintic]

In (a), you shouldn't need the algebra shown to get the straight line.
For |z| <= |z-i|, you are looking for all points which are closer to 0 than to i - it should be clear that this is the region below the line y=1/2.

In (b) the answer is an ellipse. To do this question you would need to know theory from the conics topic. Assuming you haven't done this yet, the question could not be asked yet.

 

[seanieg89]

In (a), you shouldn't need the algebra shown to get the straight line.
For |z| <= |z-i|, you are looking for all points which are closer to 0 than to i - it should be clear that this is the region below the line y=1/2.

In (b) the answer is an ellipse. To do this question you would need to know theory from the conics topic. Assuming you haven't done this yet, the question could not be asked yet.

Sure it can be asked, one can carry out the algebra without having ever heard of an ellipse before. And the graph is certainly nothing beyond what someone who has done "graphing" should be capable of.

 

[braintic]

Sure it can be asked, one can carry out the algebra without having ever heard of an ellipse before. And the graph is certainly nothing beyond what someone who has done "graphing" should be capable of.

But they haven't done Ext 2 graphing yet.
I guess it technically could be asked, but it shouldn't be.

In any case, the algebra is monstrous, requiring you to square twice to remove the square roots, which initially gives a quartic.