A Bit Of Locus Help (1 Viewer)

[Trebla]

Can someone please help me with this question:
If w = 1 - iz², find the locus of w, if Re(z) is a constant.

If I let z = x + iy, I end up with:
w = 1 - 2xy + i(y² - x²)
From there I don't know what to do next or what does Re(z) = constant supposed to do....

Any help would be greatly appreciated.........thank you

 

[SeDaTeD]

just let z = a + iy, where a is constant.

 

[YBK]

Can someone please help me with this question:
If w = 1 - iz², find the locus of w, if Re(z) is a constant.

If I let z = x + iy, I end up with:
w = 1 - 2xy + i(y² - x²)
From there I don't know what to do next or what does Re(z) = constant supposed to do....

Any help would be greatly appreciated.........thank you

-2xy should be treated like a normal number.... that's what Re(z) = constant is supposed to be.

 

[Trebla]

just let z = a + iy, where a is constant.

Ummm...can you please explain that............?

 

[SeDaTeD]

Since the real part of z is constant, only the imaginary part varies. So instead of z = x + iy, just let x=a, a constant. Also let w = u + iv, so you don't make a mistake by letting it equal x+iy. then let u = somehting, and v = something (should be clear from your workign out in your post, but just let x=a). Then write v in terms of u, and you should get a cartesian equation (should be a parabola).

 

[KeypadSDM]

-2xy should be treated like a normal number.... that's what Re(z) = constant is supposed to be.

No, no, I'm going to have to say this is both wrong and confusing.

 

[Trebla]

Thanks for the help guys, I really appreciate it...
Now, with that aside, I have another question I need help on....this time with complex roots of unity:
Let the points A1, A2,........,A11, A12 represent the 12th roots of unity, w1, w2,.......,w11, w12 and suppose P represents any complex number z such that |z|=1
............................................. _ __
i) Show that (PA1) = (z-w1)(z-w1)
ii) Prove that:
12
∑(PAn) = 24
n=1

Thanks for the help...