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 complex numbers, and 𝜃, which is greater than zero and less than or equal
to 𝜋, are constants to be found.
Remember, the locus of a point 𝑧
in this form is the arc of a circle which subtends an angle of 𝜃 between the points
represented by 𝑧 one and 𝑧 two. We have three conditions on 𝜃. If it’s less than 𝜋 by two, the
locus is a major arc. If it’s equal to 𝜋 by two, it’s a
semicircle. And if it’s greater than 𝜋 by two,
the locus is a minor arc. And, remember, the endpoints are
not part of this locus. We can see by looking at the
diagram that the locus of our 𝑧 is the major arc of a circle. And this makes sense because 𝜃 is
equal to 𝜋 by five radians.
The endpoints of our locus lie at
𝐴 and 𝐵 whose Cartesian coordinates are four, negative three and negative three,
one, respectively. These represent the complex numbers
four minus three 𝑖 and negative three plus 𝑖. And, remember, this locus is traced
in a counterclockwise direction. Since the starting point is that
represented in the complex number four minus three 𝑖, we can say that the equation
of our locus is the argument of 𝑧 minus four minus three 𝑖 over 𝑧 minus negative
three plus 𝑖 equals 𝜋 by five. We were actually told to find the
value of the constants 𝑎, 𝑏, and 𝜃. 𝑎 is four minus three 𝑖, 𝑏 is
negative three plus 𝑖, and 𝜃 is 𝜋 by five.