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Calculus/Complex numbers in argument form

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Expert: - 3/1/2009


Question
(i)two complex numbers z_1 & z_2 are given by z_1=10-2i & z_2=2-3i.Show that z_1/z_2 is of the form k(1+i) where k is real. (ii) if z=x+yi is any non-zero complex number such that z_1/z=a(1+i) where a is real,prove that 3x+2y=0.(iii)hence,find the possible values of arg z,giving your answers in degrees correct to the nearest degrees.[take arg z to be in the interval -180degree<arg z<180degree]. I only know part 1 but i'm stuck wth part 2 & 3. Please help me a bit & can you give me the final answers? Many thanks

Answer
i) Multiply the top and bottom by the conjugate of the bottom to
convert to real form.  The bottom will then be a real number and
the number can be split in two.

ii) I don't understand how to show this one.  Is it really true for
any z=x+yi?  ... Maybe I've got it.  

Take z=x+yi.  We know that (x+yi)/(10-2i) = a(1+i)
Multiply by the conjugate and you get (x+iy)/104 = (10+2i)a(1+i).
The right side is a(8+12i).  From here, it can be seen that
3*8 - 2*12 = 0, so shouldn't the equation be 3x-2y=0?

iii) Note that y/x is the tan(z), and x is known to be 2/3 of y.

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