Complex Number Question. HELP.. (1 Viewer)

[DkAssain]

let w = (3+4i)/5 and z = (5+12i)/13 so that |z|=|w|=1

(i) find wz and w(conjugate z) in the form x+iy.

(ii) hence find 2 distinct ways of writing 65^2 as the sum a^2 + b^2, where a and b are integers and 0<a<b.

having trouble with part(ii)..

 

[cutemouse]

w=(3+4i)/5
z=(5+12i)/13

wz=[(3+4i)(5+12i)]/65
=(-33+56i)/65

w(z cong)=[(3+4i)(5-12i)]/65
=(63-16i)/65

65²=33²+56²
65²=63²+16²

 

[DkAssain]

thanks

but is there an explanation on why the numerators of the solutions squared = to 65 ^2.

 

[jet]

Because |z| = |w| = 1
Then, |z||w| = 1
Then |zw| = 1
So then, calculating the mod of zw, (33^2 + 56^2/65^2) = 1 (by squaring)
<math>33^2 + 56^2 = 65^2</math>

 

[DkAssain]

ok kool thanks makes sense