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# Complex numbers p6 watch

1. Show that in an Argand diagram the equation
arg(z-2)-arg(z-2i)=3pi/4
|z-4|/|z-1| is constant on this circle.
Find the values of z corresponding to the points in which this circle is cut by the curve given by
|z-1|+|z-4|=5
2. i) arg(z-2) - arg(z-2i) = 3pi/4
arctan[y/(x-2)] - arctan[(y-2)/x] = 3pi/4

tan both sides, and you get

[y/(x-2) - (y-2)/x]/[1 + y(y-2)/x(x-2)] = -1
y/(x-2) - (y-2)/x = -1 - y(y-2)/x(x-2)

Multiply through by x(x-2) and do a bit of multiplying out and cancelling and you get x^2 + y^2 = 4, so the original equation described a circle (or an arc of it) in the Argand plane, centre (0 + 0i), radius 2.

ii) Assume that |z-4|/|z-1| is constant on this arc. If it is, then |z-4|/|z-1| = k, where k is some constant. If no value of k exists, then we know |z-4|/|z-1| is not constant on the circle.

Then |z-4| = k|z-1|
(x-4)^2 + y^2 = k^2 [(x-1)^2 + y^2]
x^2 - 8x + 16 + y^2 = k^2 [x^2 - 2x + 1 + y^2]

Now we know that x^2 + y^2 = 4 on the circle, so replace x^2 with (4 - y^2)

4 - y^2 - 8x + 16 + y^2 = k^2 [4 - y^2 - 2x + 1 + y^2]
20 - 8x = k^2 [5 - 2x]
When k^2 = 4, this equality holds. Therefore, there exists a value of k (±2), and on the circle arg(z-2) - arg(z-2i) = 3pi/4, |z-4|/|z-1| is constant. Note that the constant is actually +2, since |z-4|/|z-1| must always be positive (because of the modulus signs).

iii) |z-1| + |z-4| = 5
1 + |z-4|/|z-1| = 5/|z-1|

This curve meets the circle, and on the circle, |z-4|/|z-1| = k = +2

1 + 2 = 5/|z-1|
3|z-1| = 5
9[(x-1)^2 + y^2] = 25
25/9 = x^2 - 2x + 1 + y^2

But on the circle, x^2 = (4 - y^2)

so 25/9 = 4 - y^2 - 2x + 1 + y^2
25/9 = 5 - 2x
x = 10/9
so y = ±4.sqrt(14)/9

And the curve and circle meet at 10/9 ±4i.sqrt(14)/9
3. arctan[y/(x-2)] - arctan[(y-2)/x]

tan both sides, and you get

[y/(x-2) - (y-2)/x]/[1 + y(y-2)/x(x-2)]

This should be y/(x-2)-(y-2)/x
isnt it?
where did you get this?
1 + y(y-2)/x(x-2)
4. (Original post by totaljj)
arctan[y/(x-2)] - arctan[(y-2)/x]

tan both sides, and you get

[y/(x-2) - (y-2)/x]/[1 + y(y-2)/x(x-2)]

This should be y/(x-2)-(y-2)/x
isnt it?
where did you get this?
1 + y(y-2)/x(x-2)
tan(A-B) = tan(A)-tan(B)/1+tan(A)tan(B)
5. aha
thank you

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