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Draw an Argand diagram showing the set of points z for which the given condition is t

Draw an Argand diagram showing the set of points z for which the given condition is true.

I am working on some of these question examples and I can understand how to do if for example question is of form:

|z| = 2

or

|z - 4| <= 3

or

|z - 5j| = 6

In the above cases I am showing geometrically the difference between z and the value above. and that seems conceptually quite logical.

But then I get this:

|z + 3 - 4j| < 5

So my thinking with this one was that I would shade the circle (not including = 5 value) centred on 3 - 4i.

My reasoning for 3 - 4j as centre was that 3 - 4j + z must be < 5.

My book says centre should be -3 + 4j and I can see that technique is to change z + 3 - 4j into z - (-3 + 4j) but I can't get my head round this. Can someone please explain?

Reply 1

ghostwalker

Original post by acomber

...

As part of finding the distance between two points you subtract one from the other, so to speak.

So, |z-4| is the distance between z and 4.

Similarly |z-(-3+4j)| is the distance between z and -3+4j

Edit: If you were just working with real numbers, instead of complex, would |x+4| be obvious as the distance between x and -4 ?

(edited 9 years ago)

Reply 2

acomber

Original post by ghostwalker

OK so |z-(-3+4j)| represents the distance between z and -3+4j. I understand that. But what then does |z+(3-4j)| represent?

Reply 3

ghostwalker

Original post by acomber

OK so |z-(-3+4j)| represents the distance between z and -3+4j. I understand that. But what then does |z+(3-4j)| represent?

Mathematically they're the same.

Reply 4

acomber

Original post by ghostwalker

Mathematically they're the same.

Just been thinking about this and height of Ben Nevis + height of Snowdon does equal height Ben Nevis - (-height Snowdon). And a + b same as a - (-b). And a + bj + c + dj equal to a + bj - (-c - dj) so o I suppose yes. Confused me a bit.