Determine the ratio in which the line `3x+y=9` divides the line segment joining the points `(1,3) and (2,7)`

Answer

To determine the ratio in which the line \(3x + y = 9\) divides the line segment joining the points \((1, 3)\) and \((2, 7)\), we can follow these steps:

### Step 1: Identify the Points
Let the points be:
- \(A(1, 3)\)
- \(B(2, 7)\)

### Step 2: Assume the Ratio
Assume that the line divides the segment \(AB\) in the ratio \( \lambda : 1 \). This means we can find a point \(P\) that divides \(AB\) in this ratio.

### Step 3: Use the Section Formula
The coordinates of point \(P\) that divides the line segment joining \(A\) and \(B\) in the ratio \( \lambda : 1 \) can be calculated using the section formula:
\[ P\left( \frac{\lambda x_2 + x_1}{\lambda + 1}, \frac{\lambda y_2 + y_1}{\lambda + 1} \right) \]
Where:
- \(x_1 = 1\), \(y_1 = 3\) (coordinates of point A)
- \(x_2 = 2\), \(y_2 = 7\) (coordinates of point B)

Thus, the coordinates of point \(P\) become:
\[ P\left( \frac{\lambda \cdot 2 + 1}{\lambda + 1}, \frac{\lambda \cdot 7 + 3}{\lambda + 1} \right) \]

### Step 4: Substitute into the Line Equation
The point \(P\) must satisfy the line equation \(3x + y = 9\). Therefore, we substitute the coordinates of \(P\) into the equation:
\[ 3\left( \frac{\lambda \cdot 2 + 1}{\lambda + 1} \right) + \left( \frac{\lambda \cdot 7 + 3}{\lambda + 1} \right) = 9 \]

### Step 5: Simplify the Equation
Multiply through by \((\lambda + 1)\) to eliminate the denominator:
\[ 3(\lambda \cdot 2 + 1) + (\lambda \cdot 7 + 3) = 9(\lambda + 1) \]
This simplifies to:
\[ 6\lambda + 3 + 7\lambda + 3 = 9\lambda + 9 \]
Combining like terms gives:
\[ 13\lambda + 6 = 9\lambda + 9 \]

### Step 6: Solve for \(\lambda\)
Rearranging the equation:
\[ 13\lambda - 9\lambda = 9 - 6 \]
\[ 4\lambda = 3 \]
Thus, we find:
\[ \lambda = \frac{3}{4} \]

### Step 7: Determine the Ratio
The ratio in which the line divides the segment \(AB\) is:
\[ \lambda : 1 = \frac{3}{4} : 1 = 3 : 4 \]

### Final Answer
The line \(3x + y = 9\) divides the line segment joining the points \((1, 3)\) and \((2, 7)\) in the ratio \(3:4\).

---