Ratio in which the line 3x + 4y = 7 divides the line segment joining the points (1, 2) and (-2, 1) is
(a)`3 : 5`
(b)`4 : 6`
(c)`4 : 9`
(d)None of these

Options

A. `3 : 5`
B. `4 : 6`
C. `4 : 9`
D. None of these

Answer

To find the ratio in which the line \(3x + 4y = 7\) divides the line segment joining the points \((1, 2)\) and \((-2, 1)\), we can follow these steps:

### Step 1: Assign Coordinates
Let the points be:
- \(A(1, 2)\) as \( (x_1, y_1) \)
- \(B(-2, 1)\) as \( (x_2, y_2) \)

### Step 2: Use Section Formula
The coordinates of the point \(P\) that divides the line segment \(AB\) in the ratio \(k:1\) can be given by the section formula:
\[ P\left(\frac{kx_2 + x_1}{k + 1}, \frac{ky_2 + y_1}{k + 1}\right) \]
Substituting the coordinates of points \(A\) and \(B\):
\[ P\left(\frac{k(-2) + 1}{k + 1}, \frac{k(1) + 2}{k + 1}\right) \]

### Step 3: Set Up the Equation
The point \(P\) must satisfy the line equation \(3x + 4y = 7\). Thus, we substitute the coordinates of \(P\) into the line equation:
\[ 3\left(\frac{-2k + 1}{k + 1}\right) + 4\left(\frac{k + 2}{k + 1}\right) = 7 \]

### Step 4: Simplify the Equation
Multiply through by \(k + 1\) to eliminate the denominator:
\[ 3(-2k + 1) + 4(k + 2) = 7(k + 1) \]
Expanding this gives:
\[ -6k + 3 + 4k + 8 = 7k + 7 \]
Combining like terms results in:
\[ -2k + 11 = 7k + 7 \]

### Step 5: Solve for \(k\)
Rearranging the equation:
\[ 11 - 7 = 7k + 2k \]
\[ 4 = 9k \]
Thus,
\[ k = \frac{4}{9} \]

### Step 6: Write the Ratio
The ratio in which the line divides the segment is \(k:1\), which is:
\[ \frac{4}{9}:1 = 4:9 \]

### Conclusion
Thus, the line \(3x + 4y = 7\) divides the line segment joining the points \((1, 2)\) and \((-2, 1)\) in the ratio \(4:9\). Therefore, the correct answer is option (c) \(4:9\).