FP2: Complex numbersplex numbers

how do I do part a? I can't find the answer on examsolutions nor the textbook.

This is a standard question that you should know how to do - there will be a section on it in your textbook.

|z| is the distance from the origin to P. |z-6| is the distance from the complex number 6 ((6,0) on an argand diagram) to P. |z| = |z-6| is saying that these two distances are equal.

Which points P on the argand diagram are equidistant from (0,0) and (6,0)?

3, 0

That's one of the points but not all of them. You need the "locus of points" so you need every point that is equidistant from (0,0) and (6,0). Together these (infinite number of) points will be a line and you need to write down the equation of this line.

EDIT: Well in this case you need to draw the line instead of writing down its equation.

how would I visualise this?

Sketch an argand diagram and mark on the points (0,0) and (6,0) and then start drawing points that are the same distance from both points. E.g. (3,0) is one of them, what others are there? Together these points will form a line.

This is just GCSE maths. At GCSE you'd be asked something like, "sketch the locus of points that are equidistant from A and B".

Really you should just "know" what this locus looks like as soon as you see the equation since it's one of the standard loci. Once you've worked this out I recommend rereading through your textbook on this topic.

so this isn't related to a circle?

No not really. If it was something like |z-6|=3 then you'd draw a circle. The locus in your question is different.

ok, well that clears a lot up!

how do I do part a? I can't find the answer on examsolutions nor the textbook.

If you haven't already then I think you should watch the videos in the "Loci in the Complex Plane" section of here. You are lacking understanding.

If you haven't already then I think you should watch the videos in the "Loci in the Complex Plane" section of here. You are lacking understanding.

it says that that locus of a point that's equal distance from 2 fixed points is the perpendicular bisector...

Yes that’s right. I was trying to help you understand this without giving you the answer. If you don’t understand where these loci come from then you won’t be able to do non-standard questions.

Yes that’s right. I was trying to help you understand this without giving you the answer. If you don’t understand where these loci come from then you won’t be able to do non-standard questions.

how would I approach b? numerically..

Find the intersection of the line with the circle. Basically substitute one into the other.

Yes that’s right. I was trying to help you understand this without giving you the answer. If you don’t understand where these loci come from then you won’t be able to do non-standard questions.

Find the intersection of the line with the circle. Basically substitute one into the other.

Yes that’s right. I was trying to help you understand this without giving you the answer. If you don’t understand where these loci come from then you won’t be able to do non-standard questions.