a level further maths help!

Really stuck on this complex numbers question during my revision, goes as follows:

z1 = 2+3i
l z1z2 l = 39root2 (l z1z2 l refers to modulus of z1 * z2)
arg(z1z2) = pi/4

b) find z2, giving your answer in the form a+ib where a and b are integers (6 marks)

Any help would be greatly appreciated !

Really stuck on this complex numbers question during my revision, goes as follows:

z1 = 2+3i
l z1z2 l = 39root2 (l z1z2 l refers to modulus of z1 * z2)
arg(z1z2) = pi/4

b) find z2, giving your answer in the form a+ib where a and b are integers (6 marks)

Any help would be greatly appreciated !

There are a bunch of rules about this that can help, like:
|z1 z2| = |z1| |z2|

I think the easiest thing here is to find $z_1z_2$ in x+iy form, and then from there divide by z1 to find z2.

do you know how you would go about finding z1z2 in x+iy form?

You know the modulus and argument of z1z2. In particular, you should be able to *write down* a complex number of argument pi/2. You then just need to scale it to make the modulus match the given modulus for z1z2.

Ok so found both the modulus and argument of z2, arranged into mod-arg form, inputted the mod-arg form into my calculator and got nice integers. My final answer for a+ib was 15-3i, could you check if I'm correct, thank you in advance!

I get 15 - 3i (or 3(5 - i)) as the answer too

Did you write $z_1z_2 = re^{i\pi/4}$ where $r = |z_1z_2|$ and then divide by z_1 to find z_2? The square root of 2 cancels out nicely when you do this

Did you write $z_1z_2 = re^{i\pi/4}$ where $r = |z_1z_2|$ and then divide by z_1 to find z_2? The square root of 2 cancels out nicely when you do this

I'm not sure how much I'm being "an old fuddy duddy", but I'm somewhat despairing at the "convert everything into mod/arg form and let the calculator sort it out" mentality on display by some in this thread...

In fairness, I was a bit uneasy with this method considering the question is worth 6 marks. Do you think I would be able to achieve the full 6 marks using this?