What is the radius of a circle represented by the equation |z - z1|=k|z - z2|?

In summary, the conversation discusses the equation |z - zo|=r, representing a circle centered at a fixed point zo with radius r. It also mentions the equation |z - z1|=k|z - z2|, which also represents a circle, with a line or a circle of infinite radius as a possible result. The conversation then delves into finding the radius of the circle in this equation and presents a solution using the values of d and k.
  • #1
StephenPrivitera
363
0
In general, |z - zo|=r, where z_o is a fixed point and r is a positive number, represents a circle centered at z_o and with radius r. |z - z1|=k|z - z2|, where z_1 and z_2 are fixed points, also apparently represents a circle, except maybe in the case where k=1. Then we have a line, or a circle of infinite radius. So to find the radius of the circle for |z - z1|=k|z - z2|, I could try to rewrite the equation to fit |z - zo|=r. I did this, and I got a frightening answer. I shall attempt to show it here. The work is much too long and too tedious to write here in full form but I'll explain briefly. x is the x component of z, y is the y component of z, x_1 is the x component of z_1, y_1...y component of z_1, etc.
Square both sides, distribute the k^2, collect x's and y's on the left, complete the square to get (x-somthing)^2+(y-same thing)^2=some big mess
simplify the right hand side, rewrite in terms of z1 and z2 as much as possible, take the sqrt of each side,

Anyone who feels like trying this problem could verify my result/ show me a better way of doing it?
r=(k2-1)-1[squ][(k4-2k2+2)|z1|2+k2(2k2-1)|z2|2-2k2(x1x2+y1y2)]
 
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  • #2
Hi StephenPrivitera,
the problem is surely symmetrical wrt. the line Z1 Z2.
So let's look at this line, and the 2 points A, B where the circle intersects it (points=capital, distances=small):

Z1------ka------A----a---Z2-----b-----B
|------------------kb-----------------|

Let |Z1 - Z2| = d.
From the drawing, we see:
I. d = ka + a
II. b + d = kb
III. 2r = a+b.

Three unknowns: a, b, r.
Three equations: Perfect!
 
Last edited:
  • #3
Really there are four unknowns, a,b,r, and d. If you substitute in for d, then there are two equations and three unknowns. You can solve for r in terms of b or a. I did it for a and got,
r=(1/2)a/(k-1)

Also, how do you know that the diameter lies on the line z1z2?
 
  • #4
You know the value of d, because you know Z1 and Z2. I defined d = |Z1 - Z2|.

Concerning the symmetry: It's clear that the circle can have only 2 points in common with the line Z1 Z2. Let's assume the center C is not on Z1 Z2. Now take the mirror image C' of C wrt. to the line Z1 Z2. The circle centered at C', and going through A, B is obviously another possible solution. Now, since you stated that the circle is defined by the given equation, this is a contradiction. Thus, C is on Z1 Z2.
 
  • #5
good point, so rather than eliminating d, i should eliminate a and b;

r=dk/(k2-1)
tricky
I'm still upset the other way didn't work... There was a (k2-1)-1 factor there. Maybe the numerator can simplify somehow to dk. I'll try it again. Thanks for the help.
 
Last edited:
  • #6
Originally posted by StephenPrivitera
r=dk/(k2-1)
Correct.
Glad I could help you :wink:.
 

1. What are circles on the complex plane?

Circles on the complex plane refer to the set of points in the complex plane that are equidistant from a fixed point, called the center. They can be represented by an equation in the form of |z-z0| = r, where z is a complex number, z0 is the center, and r is the radius of the circle.

2. How are circles on the complex plane different from circles in the Cartesian plane?

Circles on the complex plane have a center and radius that are represented by complex numbers, while circles in the Cartesian plane have a center and radius that are represented by real numbers. In addition, the distance between two points on a circle in the complex plane is measured using the absolute value of complex numbers, while in the Cartesian plane it is measured using the Pythagorean theorem.

3. What is the significance of circles on the complex plane in mathematics?

Circles on the complex plane have various applications in mathematics, particularly in complex analysis. They are used to represent complex functions and to visualize transformations on the complex plane. They also play a crucial role in the study of conformal mappings and the behavior of complex functions near singularities.

4. Can circles on the complex plane intersect?

Yes, circles on the complex plane can intersect at one or more points. This occurs when the two circles have different centers and their radii are not equal. The number of intersection points can vary depending on the specific equations of the circles.

5. How do circles on the complex plane relate to polar coordinates?

Circles on the complex plane can be represented using polar coordinates, which consist of a radius and an angle. The center of the circle corresponds to the origin in polar coordinates, and the radius and angle can be used to determine the location of any point on the circle. This representation is particularly useful in complex analysis and in solving problems involving circles on the complex plane.

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