FP1 Complex

p64 FP1 Edexcel Q11


z1 = √3 + i ... z2 = 1 - i

part b) says to express 1/z1 and 1/z2 in the form a+ib
this leaves the answers
1/z1 = (√3 / 4) - ( i / 4)
1/z2 = (1/2) + (i/2)

part c) then says find the values of the real numbers of A and B such that:
A/z1 + B/z2 = z1 + z2

I'm unsure how I should approach this. I know the value of z1 + z2 will be:
1+√3 ... but that's as far as I got.


----
Q12 is also very similar. I could probably do that question if I have the right technique for this one.


Thanks in advance

right

you know the value of $z_{1} + z_{2}$

so expand out $\frac{A}{z_{1}} + \frac{B}{z_{2}}$ and equate real and imaginary parts

do you mean:

A(z2) + B(z1) / z1z2 ??

sorry what?

right, $\frac{1}{z_{1}} = \frac{\sqrt{3}-i}{4} and \frac{1}{z_{2}} = \frac{1+i}{2}$

so

$\dfrac{A(\sqrt{3}-i)}{4} + \dfrac{B(1+i)}{2} = 1 + \sqrt{3}$

now can you tidy the LHS and equate real and imaginary coefficients on the LHS and RHS?