Complex product with unknown number results in a given modulus

Question

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.

Answer 1

$$|z|=|(40-9i)(a+7i)|=|40-9i||a+7i|=41|a+7i|=1025$$ $$\therefore |a+7i|=25 \implies a=\pm 24$$

Answer 2

Well: $$(40-9i)(a+7i)=(40a+63)+(280-9a)i$$ So: $$1025^2=(40a+63)^2+(280-9a)^2$$ This is just a quadratic.