Video Transcript
The figure shows a locus of a point
𝑧 in the complex plane. Write an equation for the locus in
the form the argument of 𝑧 minus 𝑎 over 𝑧 minus 𝑏 equals 𝜃, where 𝑎 and 𝑏,
which are elements of the set of complex numbers, and 𝜃, which is greater than zero
and less than or equal to 𝜋, are constants to be found.
We recall that the locus of a point
𝑧 such that the argument of 𝑧 minus 𝑎 over 𝑧 minus 𝑏 equals 𝜃 is an arc of a
circle which subtends an angle of 𝜃 between the points represented by 𝑎 and
𝑏. It’s important that we realize that
this locus travels in a counterclockwise direction from 𝑎 to 𝑏. So for our locus, that’s this
direction. Now, if 𝜃 is less than 𝜋 by two,
the locus is a major arc. Well, we do indeed have a major
arc, but we can see that our value of 𝜃 is equal to 𝜋 by five. So that makes a lot of sense. The endpoints lie at 𝑎 and 𝑏
whose Cartesian coordinates are four, negative three and negative three, one,
respectively.
The first of these points
represents the complex number four minus three 𝑖. We know that in order to travel in
a counterclockwise direction, the locus must begin here. So we let 𝑎 be equal to four minus
three 𝑖. Similarly, it ends at the point
negative three, one, which represents the complex number negative three plus 𝑖. We can therefore say that the
equation of the locus is the argument of 𝑧 minus four minus three 𝑖 over 𝑧 minus
negative three plus 𝑖 is 𝜋 by five.
