The acute angle which the perpendicular from origin on the line `7x-3y=4` makes with the x-axis is

Options

A. zero
B. positive but not `pi//4`
C. negative
D. `pi//4`

Explanation

To find the acute angle which the perpendicular from the origin on the line \(7x - 3y = 4\) makes with the x-axis, we can follow these steps:

### Step 1: Rewrite the line equation in slope-intercept form
We start with the line equation:
\[ 7x - 3y = 4 \]
We can rearrange it to find the slope \(m\):
\[ -3y = -7x + 4 \]
\[ y = \frac{7}{3}x - \frac{4}{3} \]
From this, we can see that the slope \(m\) of the line is \(\frac{7}{3}\).

### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. Therefore, the slope \(m_p\) of the perpendicular line is:
\[ m_p = -\frac{1}{m} = -\frac{1}{\frac{7}{3}} = -\frac{3}{7} \]

### Step 3: Calculate the angle with the x-axis
The angle \(\theta\) that a line makes with the x-axis can be found using the tangent function:
\[ \tan(\theta) = m_p \]
Substituting the value of \(m_p\):
\[ \tan(\theta) = -\frac{3}{7} \]
To find the angle, we will take the arctangent:
\[ \theta = \tan^{-1}\left(-\frac{3}{7}\right) \]
Since we are interested in the acute angle, we take the absolute value:
\[ \theta = \tan^{-1}\left(\frac{3}{7}\right) \]

### Step 4: Calculate the acute angle
Using a calculator, we can find:
\[ \theta \approx 23.57^\circ \]

Thus, the acute angle which the perpendicular from the origin on the line \(7x - 3y = 4\) makes with the x-axis is approximately \(23.57^\circ\).

### Summary
The acute angle is:
\[ \theta \approx 23.57^\circ \]