Question Video: Describing Regions in the Complex Plane Mathematics

Video Transcript

The figure shows a region in the complex plane. Write an algebraic description of the shaded region.

We can clearly see that this is a circle. But there are two ways to describe the locus that forms a circle. They are the modulus of 𝑧 minus 𝑧 one equals 𝑟 and the modulus of 𝑧 minus 𝑧 one equals 𝑘 times the modulus of 𝑧 minus 𝑧 two. In this example, it makes much more sense to use the first form. In fact, we try to use this form when describing regions as it’s much more simple to find the centre and the radius then find two points whose distance to the circle are in constant ratio.

We can see that the centre of our circle is represented by the complex number four plus 𝑖. The Cartesian coordinates of this point are four, one. And we could use the distance formula to calculate the radius with either zero, seven or zero, negative five as one of the other points. Alternatively, we can find the modulus of the difference between the complex number four plus 𝑖 and either seven 𝑖 or negative five 𝑖. Let’s use seven 𝑖.

Seven 𝑖 minus four plus 𝑖 is the same as six 𝑖 minus four or negative four plus six 𝑖. So we need to find the modulus of negative four plus six 𝑖. To find the modulus, we square the real and imaginary parts, find their sum, and then find the square root of this number. So that’s the modulus of negative four squared plus six squared, which is two root 13. So we know that the boundary for our region, the circle, is described by the equation, the modulus of 𝑧 minus four plus 𝑖, cause that’s the centre, is equal to two root 13 since that’s the radius. And we can distribute these parentheses and write it as shown.

We do however need to consider the region. It’s the region outside of the circle. Each point in the region is further away from the centre of the circle than the distance of the radius. It’s also a solid line which means it represents a weak inequality. And we can therefore say that the region is represented by the modulus of 𝑧 minus four minus 𝑖 is greater than or equal to two root 13.