Video Transcript
Given that the modulus of 𝑍 plus the modulus of the complex conjugate of 𝑍 is equal to 12, determine the value of the modulus of 𝑍𝑖.
In this question, we’re given an equation involving a complex number 𝑍 and its complex conjugate. We need to use this to determine the value of the modulus of 𝑍 multiplied by 𝑖. There’s a few different ways of doing this. Let’s start by recalling some of the properties of the modulus of complex numbers. First, since we’re asked to find the modulus of 𝑍 times 𝑖 which is the product of two complex numbers, we can start by recalling the modulus of a product of two complex numbers is equal to the product of their moduli. The modulus of 𝑍 one times 𝑍 two is equal to the modulus of 𝑍 one times the modulus of 𝑍 two. We can apply this to the modulus of 𝑍 times 𝑖. It will be equal to the modulus of 𝑍 multiplied by the modulus of 𝑖.
And remember, there’s two equivalent ways of defining the modulus of a complex number. It’s the distance from the origin that that complex number lies in an Argand diagram. And this is the same as saying if the number is given in algebraic form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real numbers, then the modulus of 𝑎 plus 𝑏𝑖 is the square root of 𝑎 squared plus 𝑏 squared, the square root of the sum of the squares of the real and imaginary parts of the complex number. In an Argand diagram, 𝑖 has coordinate zero, one, so its distance from the origin is one. Therefore, the modulus of 𝑖 is equal to one. We could also see this by substituting 𝑎 is equal to zero and 𝑏 is equal to one into our formula for the modulus. The modulus of 𝑖 is the square root of one squared; it’s equal to one. Therefore, the value we’re asked to find is actually equal to the modulus of 𝑍.
To find this value, we’re going to need to use the other equation we’re given. And to do this, it might be worth noting that 𝑍 with a bar means the complex conjugate of 𝑍. This is sometimes written 𝑍 star. And the complex conjugate of 𝑍 has the same real value of 𝑍. However, its imaginary part has the opposite sign. And in particular, there’s one very useful property about complex conjugates. The modulus of the complex conjugate of 𝑍 is equal to the modulus of 𝑍. Therefore, we can substitute this into the equation we’re given, giving us that the modulus of 𝑍 plus the modulus of 𝑍 is equal to 12. We can then simplify this. We have two modulus of 𝑍 is equal to 12, and then we divide both sides of our equation through by two. We get the modulus of 𝑍 is equal to six. Finally, remember that the modulus of 𝑍 is equal to the modulus of 𝑍 times 𝑖, giving us the modulus of 𝑍 times 𝑖 is equal to six.
Therefore, given that the modulus of 𝑍 plus the modulus of the complex conjugate of 𝑍 is equal to 12, we were able to determine that the modulus of 𝑍 times 𝑖 is equal to six.
