If `m` is the slope of the straight line through the point `(1, 2)`, whose distance from the point `(13, 1)` has the greatest value, then `(2)/(3) m` is equal to

Options

A. `2`
B. `4`
C. `8`
D. `12`

Explanation

To solve the problem, we need to find the slope \( m \) of the line through the point \( (1, 2) \) that maximizes the distance to the point \( (13, 1) \). Here are the steps to arrive at the solution:

### Step 1: Understand the slope of the line
The slope \( m \) of a line through the point \( (1, 2) \) can be expressed as:
\[ m = \frac{y - 2}{x - 1} \]
where \( (x, y) \) is any point on the line.

### Step 2: Find the distance from the point \( (13, 1) \)
The distance \( d \) from the point \( (13, 1) \) to the line can be calculated using the distance formula. The distance from a point \( (x_0, y_0) \) to a line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
We need to express our line in the form \( Ax + By + C = 0 \).

### Step 3: Equation of the line
The equation of the line through \( (1, 2) \) with slope \( m \) is:
\[ y - 2 = m(x - 1) \]
Rearranging gives:
\[ mx - y + (2 - m) = 0 \]
Thus, \( A = m \), \( B = -1 \), and \( C = 2 - m \).

### Step 4: Substitute into the distance formula
Substituting \( (x_0, y_0) = (13, 1) \):
\[ d = \frac{|m(13) - 1 + (2 - m)|}{\sqrt{m^2 + 1}} \]
This simplifies to:
\[ d = \frac{|12m + 1|}{\sqrt{m^2 + 1}} \]

### Step 5: Maximize the distance
To maximize \( d \), we need to consider the expression \( |12m + 1| \). The maximum distance occurs when the line is perpendicular to the line connecting the points \( (1, 2) \) and \( (13, 1) \).

### Step 6: Calculate the slope of the line connecting the two points
The slope of the line through points \( (1, 2) \) and \( (13, 1) \) is:
\[ \text{slope} = \frac{1 - 2}{13 - 1} = \frac{-1}{12} \]

### Step 7: Find the perpendicular slope
The slope \( m \) of the line that is perpendicular to this line is the negative reciprocal:
\[ m = 12 \]

### Step 8: Calculate \( \frac{2}{3} m \)
Now we can find \( \frac{2}{3} m \):
\[ \frac{2}{3} m = \frac{2}{3} \times 12 = 8 \]

### Final Answer
Thus, the value of \( \frac{2}{3} m \) is:
\[ \boxed{8} \]