For a straight line `y=(4)/(3)x-4`. Choose correct alternate(s)

Options

A. `(dy)/(dx)=tan53^(@)`
B. `(dx)/(dy)=tan37^(@)`
C. `x-"Intercept is "3`
D. length of line between x-axis and y-axis is 5 units

Answer

To solve the problem regarding the straight line given by the equation \( y = \frac{4}{3}x - 4 \), we will analyze the equation step-by-step and determine the correctness of the provided alternatives.

### Step 1: Identify the slope of the line
The equation of the line is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

- Here, the slope \( m = \frac{4}{3} \).

### Step 2: Find the angle of inclination
The slope of the line can be related to the angle of inclination \( \theta \) using the tangent function:

\[ \tan(\theta) = \text{slope} = \frac{4}{3} \]

To find the angle \( \theta \):

\[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \]

### Step 3: Calculate the x-intercept
To find the x-intercept, set \( y = 0 \) in the equation:

\[ 0 = \frac{4}{3}x - 4 \]

Solving for \( x \):

\[ \frac{4}{3}x = 4 \implies x = 4 \cdot \frac{3}{4} = 3 \]

Thus, the x-intercept is \( 3 \).

### Step 4: Find the length of the line segment between the x-axis and y-axis
The y-intercept can be found by substituting \( x = 0 \):

\[ y = \frac{4}{3}(0) - 4 = -4 \]

The points of intersection with the axes are \( (3, 0) \) and \( (0, -4) \).

To find the length of the line segment between these two points, we can use the distance formula:

\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the points \( (3, 0) \) and \( (0, -4) \):

\[ \text{Distance} = \sqrt{(3 - 0)^2 + (0 - (-4))^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Conclusion
Based on the calculations:
1. The slope of the line is \( \frac{4}{3} \).
2. The angle of inclination \( \theta \) is \( \tan^{-1}\left(\frac{4}{3}\right) \).
3. The x-intercept is \( 3 \).
4. The length of the line segment between the x-axis and y-axis is \( 5 \) units.

Thus, all the alternatives provided in the question are correct.