To determine the equation of the line that passes through the point $T(1, -2)$ and closes an angle with the positive part of the x-axis that is twice as large as the angle closed by the line $9x - 3y + 11 = 0$, we can follow these steps:

Step 1: Find the slope of the given line.

The equation of the line is given as $9x - 3y + 11 = 0$. To find the slope, we need to express this equation in slope-intercept form, $y = mx + c$.

$9x - 3y + 11 = 0$

$-3y = -9x - 11$

$y = 3x + \frac{11}{3}$

Here, the slope $m_1$ of the given line is 3.

Step 2: Determine the angle with the x-axis.

The slope $m_1$ of a line is related to the angle $\theta_1$ that the line makes with the positive x-axis by the following relation:

$\tan(\theta_1) = m_1 = 3$

So,

$\theta_1 = \tan^{-1}(3)$

Step 3: Find the angle for the new line.

Let the angle that the required line makes with the positive x-axis be $\theta_2$. According to the problem, $\theta_2 = 2\theta_1$.

Step 4: Find the slope of the new line.

We know that:

$\tan(\theta_2) = \tan(2\theta_1)$

Using the double-angle formula for tangent:

$\tan(2\theta_1) = \frac{2\tan(\theta_1)}{1 - \tan^2(\theta_1)}$

Substituting $\tan(\theta_1) = 3$:

$\tan(2\theta_1) = \frac{2 \times 3}{1 - 3^2} = \frac{6}{1 - 9} = \frac{6}{-8} = -\frac{3}{4}$

So, the slope $m_2$ of the required line is $-\frac{3}{4}$.

Step 5: Use the point-slope form to find the equation of the line.

The line passes through the point $T(1, -2)$ and has a slope $m_2 = -\frac{3}{4}$. The point-slope form of the equation of a line is given by:

$y - y_1 = m(x - x_1)$

Substituting $(x_1, y_1) = (1, -2)$ and $m = -\frac{3}{4}$:

$y + 2 = -\frac{3}{4}(x - 1)$

Multiply both sides by 4 to eliminate the fraction:

$4(y + 2) = -3(x - 1)$

$4y + 8 = -3x + 3$

$3x + 4y = -5$

So, the equation of the required line is:

$\boxed{3x + 4y + 5 = 0}$

Would you like any details or have any questions?

Here are 5 related questions for further exploration:

1. How can the angle of inclination of a line be derived from its slope?
2. What is the geometric interpretation of the tangent function in the context of a line’s slope?
3. How do different transformations affect the slope of a line?
4. How can the equation of a line be determined if two points on the line are known?
5. How would the problem change if the angle was three times as large instead of twice?

Tip: When working with slopes, remember that positive slopes indicate lines that rise as they move from left to right, while negative slopes indicate lines that fall.