Loci Defined by Differences of Arguments
The equation
defines a locus of points. In fact it defines part of a circle. We can write
so that the locus is the set of points
with a difference in the angle with the real axis of
between the lines drawn from the points
and
torespectively.
The argument offrom
is
and the argument offrom
is
so that
We can find the angle formed atby the lines to -1 and 2 on the real axis in terms ofand
Draw a vertical line fromand label angles as shown below.
Fromdrop a vertical and label the angles formed as below.
Atthe two angles
and
add to give angles
which equalsfrom the original condition. This means that the interval from -1 to 2 is the diameter of a circle andis on the circumference, since the angle subtended equals
The centre of the circle is
and the radius is 1.5. The equation of the circle is
