Centre of a circle & complex numbers

In summary, the conversation discusses finding the locus of z and the centre of the circle with the given equation. It is mentioned that the locus has been sketched but the centre of the circle cannot be determined algebraically. The speaker also asks for assistance in finding at least three points on the circle and suggests a geometric interpretation to determine the angle between 0 and 2. The importance of symmetry in the problem is also mentioned.
  • #1
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[tex] arg(\dfrac{z}{z-2}) = \dfrac{\pi}{3} [/tex]

sketch the locus of z and find the centre of the circle

I've sketched the locus of z but I can't seem to find the centre of the circle. Is there a way to do it algebraically? I've attempted to use z = x + iy, but to no avail.
 
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  • #2
Can you find some (at least 3) points on the circle? The points with minimal/maximal real or imaginary part would be ideal.

A geometric interpretation (for all points, what is the apparent angle between 0 and 2?) is nice, too.

And if you see the symmetry of the problem, it is even better.
 

1. What is the formula for finding the centre of a circle?

The centre of a circle can be found by taking the average coordinates of the endpoints of the diameter or by using the formula (h,k), where h is the x-coordinate and k is the y-coordinate of the centre.

2. How do complex numbers relate to the centre of a circle?

Complex numbers can be used to represent points on a 2-dimensional plane, making them useful for finding the centre of a circle. The real part of a complex number represents the x-coordinate, while the imaginary part represents the y-coordinate.

3. Can the centre of a circle be located outside of the circle?

No, the centre of a circle must always lie inside the circle. This is because the centre is equidistant from all points on the circumference of the circle.

4. How do you find the centre of a circle with complex numbers?

To find the centre of a circle using complex numbers, you can use the formula (h,k) where h is the average of the real parts of the complex numbers representing the endpoints of the diameter, and k is the average of the imaginary parts.

5. Can the centre of a circle be a complex number?

Yes, the centre of a circle can be a complex number. This is because complex numbers can represent any point on a 2-dimensional plane, including the centre of a circle.

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