A line with positive rational slope, passes through the point A(6,0) and is at a distance of 5 units from B (1,3). The slope of line is

Options

A. `(15)/(8)`
B. `(8)/(15)`
C. `(5)/(8)`
D. `(8)/(5)`

Explanation

To solve the problem, we need to find the slope \( m \) of a line that passes through the point \( A(6, 0) \) and is at a distance of 5 units from the point \( B(1, 3) \). The slope \( m \) must be positive and rational.

### Step-by-step Solution:

1. **Identify the point and slope**:
The line passes through point \( A(6, 0) \). Let the slope of the line be \( m \). The equation of the line in point-slope form is given by:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (6, 0) \):
\[ y - 0 = m(x - 6) \implies y = mx - 6m \]

2. **Rewrite the equation in standard form**:
Rearranging the equation gives:
\[ mx - y - 6m = 0 \]
Here, \( a = m \), \( b = -1 \), and \( c = -6m \).

3. **Use the distance formula**:
The distance \( d \) from a point \( (x_0, y_0) \) to a line \( ax + by + c = 0 \) is given by:
\[ d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}} \]
For point \( B(1, 3) \), we substitute \( (x_0, y_0) = (1, 3) \):
\[ d = \frac{|m(1) + (-1)(3) - 6m|}{\sqrt{m^2 + (-1)^2}} = \frac{|m - 3 - 6m|}{\sqrt{m^2 + 1}} = \frac{| -5m - 3|}{\sqrt{m^2 + 1}} \]

4. **Set the distance equal to 5**:
According to the problem, this distance is equal to 5:
\[ \frac{| -5m - 3|}{\sqrt{m^2 + 1}} = 5 \]

5. **Cross-multiply to eliminate the fraction**:
\[ | -5m - 3| = 5\sqrt{m^2 + 1} \]

6. **Square both sides**:
To eliminate the absolute value, we consider two cases:
- Case 1: \( -5m - 3 = 5\sqrt{m^2 + 1} \)
- Case 2: \( -5m - 3 = -5\sqrt{m^2 + 1} \)

Let's solve Case 1:
\[ (-5m - 3)^2 = (5\sqrt{m^2 + 1})^2 \]
This simplifies to:
\[ 25m^2 + 30m + 9 = 25(m^2 + 1) \]
\[ 25m^2 + 30m + 9 = 25m^2 + 25 \]
\[ 30m + 9 - 25 = 0 \implies 30m - 16 = 0 \implies m = \frac{16}{30} = \frac{8}{15} \]

7. **Verify the slope**:
Since \( m = \frac{8}{15} \) is positive and rational, it satisfies the conditions of the problem.

### Final Answer:
The slope of the line is:
\[ m = \frac{8}{15} \]