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# complex numbers watch

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1. a = 1 +4j
a^2 = 8j - 15
a^3 = -52j - 47

a satisfies

z^3 + 5z^2 +kz +m = 0

find k and show m = 119

ive dun questions like this before but just forgot how to go about it - thanx

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i think i got it by comparing coefficients

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so dont worry bout it - thanx
2. they then ask for the 2 other roots of the equation. So there is the conjugate 1-4j and i got -7 as the other root. it then asks for their arguments and i got 0 for -7 and +/- 1.33 for the other two roots respectively.

the next part is :

verify that there is a constant c such that all three roots satisfy |z+2| = c

draw an argand diagram showing the locus of points representing all complex numbers z for which |z+2| = c.

ne idea?
3. c = 5, which you can see by plugging the roots into |z + 2|.

|z + 2| = 5 is a circle with its center at (-2, 0) and radius 5.
4. as dvs says...I was just about to write the same sort of thing
5. (Original post by dvs)
c = 5, which you can see by plugging the roots into |z + 2|.

|z + 2| = 5 is a circle with its center at (-2, 0) and radius 5.
cool thanx - but how did u no the centre?

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its ok i see now
6. For a specific locus, |z-(a+bj)| = r, this is the same as saying 'where can the points of z lie, such that a vector from (a+bj) to z has a distance r.

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Updated: December 5, 2005
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