The line segment joining the points A(3,2) and B (5,1) is divided at the point P in the ratio `1 : 2` and it lies on the line `3x-18y+k=0`. Find the value of k.

Options

A. `19`
B. `18`
C. `17`
D. `16`

Explanation

To solve the problem step by step, we will find the coordinates of point P that divides the line segment AB in the ratio 1:2, and then we will substitute these coordinates into the line equation to find the value of k.

### Step 1: Identify the coordinates of points A and B
Given:
- Point A (x1, y1) = (3, 2)
- Point B (x2, y2) = (5, 1)

### Step 2: Use the section formula to find the coordinates of point P
The section formula states that if a point P divides the line segment joining points A (x1, y1) and B (x2, y2) in the ratio m:n, then the coordinates of P are given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Here, m = 1 and n = 2.

Substituting the values:
\[ P\left(\frac{1 \cdot 5 + 2 \cdot 3}{1 + 2}, \frac{1 \cdot 1 + 2 \cdot 2}{1 + 2}\right) \]

Calculating the x-coordinate:
\[ P_x = \frac{5 + 6}{3} = \frac{11}{3} \]

Calculating the y-coordinate:
\[ P_y = \frac{1 + 4}{3} = \frac{5}{3} \]

Thus, the coordinates of point P are:
\[ P\left(\frac{11}{3}, \frac{5}{3}\right) \]

### Step 3: Substitute the coordinates of P into the line equation
The line equation is given by:
\[ 3x - 18y + k = 0 \]
Substituting \(x = \frac{11}{3}\) and \(y = \frac{5}{3}\):
\[ 3\left(\frac{11}{3}\right) - 18\left(\frac{5}{3}\right) + k = 0 \]

Calculating:
\[ 11 - 30 + k = 0 \]
\[ -19 + k = 0 \]

### Step 4: Solve for k
\[ k = 19 \]

### Final Answer
The value of k is:
\[ \boxed{19} \]