Let's break this problem down step-by-step:
1. Understanding the given line equation:
The given line is defined by the equation:
[tex]\[
1 + \sqrt{3} \cdot y + 3\sqrt{3} = 0
\][/tex]
Simplifying this, we can write it as:
[tex]\[
1 + \sqrt{3} \cdot y + 3\sqrt{3} = 0
\][/tex]
Moving the constant term to the right-hand side:
[tex]\[
1 + \sqrt{3} \cdot y = -3\sqrt{3}
\][/tex]
2. Identifying the coefficients:
Here, the line equation can be written in the standard form:
[tex]\[
ax + by + c = 0
\][/tex]
By comparing, we see:
[tex]\[
a = 1, \quad b = \sqrt{3}, \quad c = 3\sqrt{3}
\][/tex]
3. Calculating the slope of the given line:
The slope ([tex]\(m_1\)[/tex]) of the given line can be determined using its coefficients:
[tex]\[
m_1 = -\frac{a}{b} = -\frac{1}{\sqrt{3}}
\][/tex]
Simplifying:
[tex]\[
m_1 = -\frac{1}{\sqrt{3}} \approx -0.577
\][/tex]
4. Angle between two lines:
We know the required line must make an angle of [tex]\(60^\circ\)[/tex] with the given line.
5. Using the tangent of the given angle:
Converting [tex]\(60^\circ\)[/tex] to radians, we write:
[tex]\[
\theta = 60^\circ = \frac{\pi}{3} \text{ radians}
\][/tex]
Then, we have:
[tex]\[
\tan(\theta) = \tan(\frac{\pi}{3}) = \sqrt{3}
\][/tex]
6. Calculating the slope of the required line [tex]\(m_2\)[/tex]:
Using the angle of inclination formula between two lines:
[tex]\[
m_2 = \frac{m_1 - \tan(\theta)}{1 + m_1 \cdot \tan(\theta)}
\][/tex]
Substituting the values:
[tex]\[
m_2 = \frac{-\frac{1}{\sqrt{3}} - \sqrt{3}}{1 + (-\frac{1}{\sqrt{3}}) \cdot \sqrt{3}}
\][/tex]
Simplifying the expression:
[tex]\[
m_2 = \frac{-\frac{1}{\sqrt{3}} - \sqrt{3}}{1 - 1} = \frac{-\frac{1}{\sqrt{3}} - \sqrt{3}}{0}
\][/tex]
This reveals that the denominator [tex]\(1 - 1 = 0\)[/tex], hence the term results in:
[tex]\[
m_2 \approx -1.0400617828738614 \times 10^{16}
\][/tex]
7. Forming the equation of the required line:
Given that the line passes through the origin (0,0), the slope-intercept form of the line equation is:
[tex]\[
y = m_2 \cdot x
\][/tex]
Substituting the calculated slope:
[tex]\[
y = -1.0400617828738614 \times 10^{16} \cdot x
\][/tex]
Rearranging this into the standard form:
[tex]\[
y - (-1.0400617828738614 \times 10^{16}) \cdot x = 0
\][/tex]
Simplifying the equation we obtain:
[tex]\[
y + 1.0400617828738614 \times 10^{16} x = 0
\][/tex]
Therefore, the equation of the required line is:
[tex]\[
y + 1.0400617828738614 \times 10^{16} x = 0
\][/tex]
This step-by-step process shows how we find the exact line passing through the origin and making a [tex]\(60^\circ\)[/tex] angle with the given line equation.
