Question Video: Locus Defined Using the Argument of a Quotient of Two Complex Numbers

The point 𝑧 satisfies arg ((𝑧 βˆ’ 6)/(𝑧 βˆ’ 6𝑖)) = πœ‹/4. Plot the locus of 𝑧 on an Argand diagram.

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Video Transcript

The point 𝑧 satisfies the equation the argument of 𝑧 minus six over 𝑧 minus six 𝑖 equals πœ‹ by four. Plot the locus of 𝑧 on an Argand diagram.

The locus of 𝑧 is the arc of a circle which subtends an angle of πœ‹ by four radians between the points represented by six and six 𝑖. Remember, these are plotted in a counterclockwise direction from six to six 𝑖. But they don’t 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œ‹ by four is less than πœ‹ 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, it’s the major arc of a circle. But without knowing the centre of the circle, we can’t 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. It’s not actually necessary to add the cords shown. But by doing so, we can see that we will get an angle that’s less than πœ‹ 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.

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