WEBVTT
00:00:00.534 --> 00:00:02.714
The figure shows a region in the complex plane.
00:00:03.004 --> 00:00:05.904
Write an algebraic description of the shaded region.
00:00:06.474 --> 00:00:08.654
We can clearly see that this is a circle.
00:00:08.944 --> 00:00:13.224
But there are two ways to describe the locus that forms a circle.
00:00:13.614 --> 00:00:21.314
They are the modulus of π§ minus π§ one equals π and the modulus of π§ minus π§ one equals π times the modulus of π§ minus π§ two.
00:00:21.924 --> 00:00:25.454
In this example, it makes much more sense to use the first form.
00:00:25.894 --> 00:00:35.504
In fact, we try to use this form when describing regions as itβs much more simple to find the centre and the radius then find two points whose distance to the circle are in constant ratio.
00:00:36.134 --> 00:00:40.744
We can see that the centre of our circle is represented by the complex number four plus π.
00:00:41.184 --> 00:00:44.124
The Cartesian coordinates of this point are four, one.
00:00:44.954 --> 00:00:52.484
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.
00:00:53.084 --> 00:01:00.114
Alternatively, we can find the modulus of the difference between the complex number four plus π and either seven π or negative five π.
00:01:00.594 --> 00:01:01.734
Letβs use seven π.
00:01:02.214 --> 00:01:08.304
Seven π minus four plus π is the same as six π minus four or negative four plus six π.
00:01:08.474 --> 00:01:11.234
So we need to find the modulus of negative four plus six π.
00:01:11.694 --> 00:01:18.864
To find the modulus, we square the real and imaginary parts, find their sum, and then find the square root of this number.
00:01:19.134 --> 00:01:24.054
So thatβs the modulus of negative four squared plus six squared, which is two root 13.
00:01:24.564 --> 00:01:35.934
So we know that the boundary for our region, the circle, is described by the equation, the modulus of π§ minus four plus π, cause thatβs the centre, is equal to two root 13 since thatβs the radius.
00:01:36.264 --> 00:01:38.954
And we can distribute these parentheses and write it as shown.
00:01:39.454 --> 00:01:41.714
We do however need to consider the region.
00:01:42.014 --> 00:01:44.604
Itβs the region outside of the circle.
00:01:44.974 --> 00:01:50.954
Each point in the region is further away from the centre of the circle than the distance of the radius.
00:01:51.324 --> 00:01:54.994
Itβs also a solid line which means it represents a weak inequality.
00:01:55.354 --> 00:02:03.884
And we can therefore say that the region is represented by the modulus of π§ minus four minus π is greater than or equal to two root 13.