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.