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.

Explanation

To solve the problem, we need to find the value of \( k \) such that the point \( P \), which divides the line segment joining points \( A(3,2) \) and \( B(5,1) \) in the ratio \( 1:2 \), lies on the line given by the equation \( 3x - 18y + k = 0 \).

### Step 1: Use the Section Formula
The coordinates of point \( P \) that divides the line segment \( AB \) in the ratio \( m:n = 1:2 \) can be calculated using the section formula:

\[ P(x,y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
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