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Chapter 2: Problem 135

Let \(z_{1}\) and \(z_{2}\) be two roots of equation \(z^{2}+a z+b=0 . Z\) is complex number. Assume that origin, \(\mathrm{z}_{1}\) and \(\mathrm{z}_{2}\) from an equilateral triangle then (a) \(\mathrm{a}^{2}=2 \mathrm{~b}\) (b) \(\mathrm{a}^{2}=3 \mathrm{~b}\) (c) \(a^{2}=4 b\) (b) \(a^{2}=b\)

Short Answer

Expert verified
\( a^2 = 2b \)

Step by step solution

01

Write down the relationship between the roots and the coefficients of the equation

We are given that \(z_1\) and \(z_2\) are roots of the equation \(z^2 + az + b = 0\). By Vieta's formulas, we have the following relationships between the roots and the coefficients: \( z_1 + z_2 = -a \) \( z_1 z_2 = b \) We will use these relationships later in our solution.
02

Find the difference between the complex numbers

Since the origin forms one vertex of the equilateral triangle, we can consider the complex numbers \(z_1\) and \(z_2\) as the other two vertices. Now let's find the difference between the two complex numbers: \( z_2 - z_1 \) We'll use this difference later when considering the geometric properties of the equilateral triangle.
03

Determine the properties of the equilateral triangle.

In an equilateral triangle, all sides have equal length and internal angles are all equal to 60 degrees. In terms of complex numbers, this means that the magnitude of the difference between any two vertices is equal, and the angle between the vertices is a multiple of \( 60^{\circ} \), or \(\frac{\pi}{3} \) radians. Since the magnitudes of the differences between the vertices are equal, we have the following relationship: \( |z_2 - z_1| = |z_1| = |z_2| \) Also, the angles between the vertices can be expressed as follows: \( \text{arg}\, (z_2 - z_1) = \text{arg}\, z_1 + \frac{\pi}{3} \) or \( \text{arg}\, (z_2 - z_1) = \text{arg}\, z_1 - \frac{\pi}{3} \)
04

Express the differences using the properties of the equilateral triangle

We can express the difference between the complex numbers using polar form: \( z_2 - z_1 = r(\cos(\text{arg} \, z_1 \pm \frac{\pi}{3}) + i \sin(\text{arg} \, z_1 \pm \frac{\pi}{3})) \) where \(r = |z_1| = |z_2|\) is the distance between vertices, and the plus and minus signs indicate the two different possibilities for the angle between vertices, as explained in the previous step.
05

Express the sum using the properties of the equilateral triangle and Vieta's formulas

Now we will take the conjugate of both sides of the equation above: \( \overline{z_2 - z_1} = r(\cos(-\text{arg} \, z_1 \mp \frac{\pi}{3}) + i \sin(-\text{arg} \, z_1 \mp \frac{\pi}{3})) \) Adding the two equations, we get: \( z_1 + z_2 = r(\cos(\text{arg} \, z_1 \pm \frac{\pi}{3}) + \cos(-\text{arg} \, z_1 \mp \frac{\pi}{3}) + i (\sin(\text{arg} \, z_1 \pm \frac{\pi}{3}) + \sin(-\text{arg} \, z_1 \mp \frac{\pi}{3}))) \) Now we can use the fact that \( z_1 + z_2 = -a \) from Vieta's formulas, \( -a = r(2\cos(\text{arg} \, z_1 \pm \frac{\pi}{3})) \)
06

Express the product using the properties of the equilateral triangle and Vieta's formulas

Similarly, take the product of both sides of the equation in Step 4: \( (z_2 - z_1)(\overline{z_2 - z_1}) = r^2(1 + 2\cos(2(\text{arg} \, z_1 \pm \frac{\pi}{3})) \) Using the property that \( z_1 z_2 = b \) from Vieta's formulas: \( -b = r^2(1 + 2\cos(2(\text{arg} \, z_1 \pm \frac{\pi}{3})) \)
07

Use trigonometric identities and the results from Step 5 and 6 to find the relationship between a and b

Using the trigonometric identity \( \cos(2\theta) = 2\cos^2(\theta) - 1 \), we get: \( -b = r^2(1 + 2(2\cos^2(\text{arg} \, z_1 \pm \frac{\pi}{3}) - 1)) \) Using the relationship \( -a = r(2\cos(\text{arg} \, z_1 \pm \frac{\pi}{3})) \) from Step 5, we have: \( -b = \frac{(-a)^2}{4}(2 - 2\cos^2(\text{arg} \, z_1 \pm \frac{\pi}{3})) \) Finally, using the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) and simplifying, we obtain the relationship between \(a\) and \(b\): (a) \( a^2 = 2b \) Thus, when the complex numbers \(0\), \(z_1\), and \(z_2\) form an equilateral triangle, the relationship between the coefficients \(a\) and \(b\) is given by \(a^2 = 2b\).

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