Video Transcript
The point π§ satisfies the equation
the argument of π§ minus six over π§ minus six π equals π by four. Plot the locus of π§ on an Argand
diagram.
The locus of π§ is the arc of a
circle which subtends an angle of π by four radians between the points represented
by six and six π. Remember, these are plotted in a
counterclockwise direction from six to six π. But they donβt actually include
these points themselves. These are the points on the Argand
plane whose Cartesian coordinates are six, zero and zero, six, respectively. And since π by four is less than
π by two, we know that we have a major arc. So we begin by adding the points
six, zero and zero, six on our Argand diagram.
And then we come across a
problem. How do we know where the major arc
sits? Sure, itβs the major arc of a
circle. But without knowing the centre of
the circle, we canβt use that information to find the arc. It could actually be either of
these two arcs shown. Here, we recall the fact that the
locus is drawn in a counterclockwise direction. We need the arc that begins at the
point six, zero and ends up at the point zero, six to be a major arc, when drawn in
this direction.
This means we have to choose this
arc on the right. And, therefore, the locus is as
shown. Itβs not actually necessary to add
the cords shown. But by doing so, we can see that we
will get an angle thatβs less than π by two radians. It is also possible to find the
Cartesian equation of loci in this form. Occasionally, you can take a
geometric approach. But, in general, an algebraic
approach is sensible.
