# Complex product with unknown number results in a given modulus

I have the following complex number $$z = (40-9i)·(a+7i)$$ and I want to find the number $$a$$ such that $$\left| z \right| = 1025$$

I tried to solve it through a couple of methods such as developing $$z$$ to represent the product, try to isolate $$a$$, try to put the modulus formula in an equation-like fashion but I'm stuck in the same place.

$$|z|=|(40-9i)(a+7i)|=|40-9i||a+7i|=41|a+7i|=1025$$ $$\therefore |a+7i|=25 \implies a=\pm 24$$
• In "polar form" we can write the two numbers as $a_1e^{i\theta}_1$ and $a_2e^{\theta_2}$ where $a_1$ and $a_2$ are the moduli of the two complex numbers and $\theta_1$ and $\theta_2$ are the "arguments" so that the product is $(a_1a_2)e^{i(\theta_1+ \theta_2)}$. That clearly has modulus $a_1a_2$. That is why Peter Foreman can just multiply the two moduli. Mar 10 '19 at 16:03
Well: $$(40-9i)(a+7i)=(40a+63)+(280-9a)i$$ So: $$1025^2=(40a+63)^2+(280-9a)^2$$ This is just a quadratic.