complex number problem (1 Viewer)

[stag_j]

this question was in my trial exam and i have an answer for it but don't quite understand the process.

draw a clear sketch to show the important features of the curve defined by: |Z-Z1| + |Z-Z2| = 12
where Z1=3+4i and Z2=9+4i

the answer is an ellipse, with foci at (3,4) and (9,4).
i can see that this is probably similar to the simpler questions like |Z-Z1| = 5 but i'm a bit stuck for when theres two points...

 

[turtle_2468]

Hmm... well sort of.
Consider the points Z, Z_1, Z_2 on an argand diagram. Then the equation given becomes:
The distance from Z to Z_1 PLUS the distance from Z to Z_2 is 12.
Which, if you remember, is a way of defining an ellipse!
Thus it's an ellipse with foci at the given points.

 

[deyveed]

Yeah
I'm having trouble with these type of questions too.
Could you tell me how to answer these sort of questions (not this but similar ones) and the Arg[(z - z_1)/(z - z_2)] ones without having to memorise that |z - z_1| = c is a circle with z_1 centre etc.
Just some sort of general way to work them out (algebraicly, graphically etc)

 

[stag_j]

can someone tell me the equation of the ellipse at least?
maybe then i could work backwards.

 

[deyveed]

You mean x^2/a^2 + y^2/b^2 = 1 ?

 

[stag_j]

no i meant the equation of the ellipse described in this question. i'm assuming it would be something along the lines of (x-6)^2/a^2 + (y-4)^2/b^2=1
except i dont know what a and b are in this.

 

[deyveed]

I think its (x-6)^2/9 + (y-4)^2/27 = 1
Not 100% though

 

[ND]

Originally posted by stag_j
no i meant the equation of the ellipse described in this question. i'm assuming it would be something along the lines of (x-6)^2/a^2 + (y-4)^2/b^2=1
except i dont know what a and b are in this.

Well you know what the sum of the SP and S'P is, so just think about where P is when S'SP is a line for a, and when SP=S'P for b.