The picture below shows a figure in the complex plane, consisting of two circles of radius 1, with centers (3,0) and (−3,0) and a lower half of the circle of radius 2 with the center (0,−23√).
Find the number of positive integers n≤1000, such that for some complex number a the equation zn−a=0 has a root on each of the three parts of the figure.
Details and assumptions
The three parts of the figure the question refers to are the two circles and the semi-circle.
To find the number of positive integers n ≤ 1000 such that the equation zn − a = 0 has a root on each of the three parts of the figure, we need to analyze the behavior of the equation for different values of n.
Let's start by considering the roots on the two circles. We know that the circles have a radius of 1 and their centers are (3,0) and (-3,0) respectively. Therefore, the equations of the two circles can be written as:
Circle 1: (x - 3)^2 + y^2 = 1
Circle 2: (x + 3)^2 + y^2 = 1
Now, let's consider the roots on the lower half of the circle. The radius of this circle is 2 and its center is (0, -2√3). The equation of this circle can be written as:
Circle 3: x^2 + (y + 2√3)^2 = 4
To find the roots that satisfy the equation zn − a = 0, we can equate the magnitude of the complex number zn to the magnitude of a. Let's consider each part of the figure separately.
1. Circle 1:
For the roots on Circle 1, let's write the complex number zn as re^(θi), where r is the magnitude (distance from the origin to the point) and θ is the angle from the positive x-axis. Setting the magnitude of zn equal to the magnitude of a, we get:
|re^(θi)| = |a|
|r| = |a|
Since we're considering the positive integers n, let's assume n = k for some positive integer k. For k = 1, the root on Circle 1 will be (1,0). This satisfies |a| = 1. For other values of k, the roots on Circle 1 will be k equally spaced points. Since we want a root on Circle 1, we need to make sure these points lie on the circle. Therefore, we can say that:
|re^(θi)| = |a| = 1
r = 1
So, for any positive integer k, the equation zn − a = 0 will have roots on Circle 1.
2. Circle 2:
Similar to Circle 1, for the roots on Circle 2, let's write the complex number zn as re^(θi) and set the magnitude of zn equal to the magnitude of a:
|re^(θi)| = |a|
|r| = |a|
Using the same reasoning as before, we find that for any positive integer k, the equation zn − a = 0 will have roots on Circle 2.
3. Circle 3:
For the roots on the lower half of Circle 3, let's consider the complex number zn as re^(θi) again and set the magnitude of zn equal to the magnitude of a:
|re^(θi)| = |a|
|r| = |a|
Since we're considering the lower half of Circle 3, we need to make sure that the imaginary part of zn is negative. This can be achieved by setting the argument (angle from the positive x-axis) θ to lie between -2π/3 and 2π/3. Therefore, we can say that:
-2π/3 ≤ θ ≤ 2π/3
For any positive integer k, the equation zn − a = 0 will have roots on the lower half of Circle 3.
From the above analysis, we can see that for any positive integer n and any complex number a, the equation zn − a = 0 will have a root on each of the three parts of the figure. Therefore, the number of positive integers n ≤ 1000 satisfying the given condition is 1000.
To solve this problem, we need to find the number of positive integers n≤1000 for which the equation zn−a=0 has a root on each of the three parts of the given figure.
Let's break down the problem into three parts: the two circles and the semi-circle.
1. The circle with center (3,0) and radius 1:
The equation of this circle can be expressed as (x - 3)^2 + y^2 = 1.
To find the roots, zn − a = 0, on this circle, we substitute x + yi into the equation and solve for the value of n.
We need to find integers n for which the equation has a solution on this circle.
2. The circle with center (-3,0) and radius 1:
The equation of this circle can be expressed as (x + 3)^2 + y^2 = 1.
Similarly, substitute x + yi into the equation zn − a = 0 and solve for the value of n.
We need to find integers n for which the equation has a solution on this circle.
3. The lower half of the circle with center (0, -2√3) and radius 2:
The equation of this circle can be expressed as x^2 + (y + 2√3)^2 = 4.
Once again, substitute x + yi into the equation zn − a = 0 and solve for the value of n.
We need to find integers n for which the equation has a solution on this lower half of the circle.
Now, to find the total number of positive integers n≤1000 that satisfy the given conditions, we need to count the number of common roots on all three parts of the figure.
One way to do this is to find the values of n for which the equations in each part of the figure have common roots.
You can write a program or use a computer algebra system (such as Mathematica) to solve the equations zn − a = 0 on each part of the figure and identify the common roots.
By iterating through the values of n from 1 to 1000 and counting the number of common roots obtained, you can determine the total number of positive integers n≤1000 that satisfy the given conditions.
Remember to handle complex solutions properly and consider both real and complex roots.
Therefore, by finding the common roots on each part of the figure and counting them, you can determine the number of positive integers n≤1000 for which the equation zn − a = 0 has roots on each part of the given figure.