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# FP2 Loci watch

1. Please could someone explain part a) and part c) to me?

For a) I understand that pi/2 implies a semi-circle, but could someone show from scratch how they would approach this question?

Also, please could someone explain what's going in the graph for part c), because it looks nothing like a half line to me!

Thanks
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2. if you let z = x + iy

(x + iy - 2i)/(x + iy + 2) can be multiplied by (x + 2 - iy)/(x + 2 - iy) and simplified to ax + iby.

the arg of (p + iq) = pi/2, so tan pi/2 = q/p = tan (pi/2)

so since tan ( pi/2 ) is undefined it follows that p = 0... from there you get the equation of a circle.
3. (Original post by the bear)
if you let z = x + iy

(x + iy - 2i)/(x + iy + 2) can be multiplied by (x + 2 - iy)/(x + 2 - iy) and simplified to ax + iby.

the arg of (p + iq) = pi/2, so tan pi/2 = q/p = tan (pi/2)

so since tan ( pi/2 ) is undefined it follows that p = 0... from there you get the equation of a circle.

I'm sorry but I still don't understand. I have no idea how to construct an answer to the question. I.e. I'm not sure why the semi-circle is in the quadrant it's in.
4. (Original post by PhyM23)
Zacken
You know that the is given by (with some restrictions that we'll talk about later) we then have:

, rationalise this - the denominator will be .

But since you know: (unrigorously) where the rationalised thing.

But the only way you can have something be infinity is if the denominator is zero. So you know that which is the equation of a circle.
5. (Original post by Zacken)
You know that the is given by (with some restrictions that we'll talk about later) we then have:

, rationalise this - the denominator will be .

But since you know: (unrigorously) where the rationalised thing.

But the only way you can have something be infinity is if the denominator is zero. So you know that which is the equation of a circle.

That makes sense, but when I expand I get (x+2)^2 + y^2 = 0, but the radius can't be zero?

And how does finding the equation of the circle help with knowing which quadrant to put the semi-circle in, and where you have to draw the half-lines to show where the angles make pi/2?
6. (Original post by PhyM23)
...
Lol. ignore what I said. We have , rationalise this and then separate it into real and imaginary parts. Let's call the imaginary part and the real part .

Then you know that . This will get you the equation of a circle.

Now you know that you need - so that gets you the restrictions that forces it to be only a semi-circle.
7. (Original post by Zacken)
Lol. ignore what I said. We have , rationalise this and then separate it into real and imaginary parts. Let's call the imaginary part and the real part .

Then you know that . This will get you the equation of a circle.

Now you know that you need - so that gets you the restrictions that forces it to be only a semi-circle.

Ah I think I understand part a) now, except I'm not sure how gets the restrictions for a semi-circle...

Also, why have they in the solutions I posted not just done a single half line for part c)? The graph looks like a mess!

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