Segment joining (1,2) and (-2,1) is divided by the line 3x+4y=7 in the ration

Options

A. `3:4`
B. `4:3`
C. `9:4`
D. `4:9`

Answer

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

### Step 1: Define the Points and the Line
We have two points:
- \(A(1, 2)\)
- \(B(-2, 1)\)

And the line equation is:
\[ 3x + 4y = 7 \]

### Step 2: Set Up the Ratio
Let the point \(Q\) divide the segment \(AB\) in the ratio \(\lambda : 1\). The coordinates of point \(Q\) can be found using the section formula:
\[ Q\left(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\right) \]
where \(m_1 = \lambda\) and \(m_2 = 1\).

### Step 3: Calculate the Coordinates of \(Q\)
Using the section formula, the coordinates of \(Q\) are:
\[ x_Q = \frac{\lambda \cdot (-2) + 1 \cdot 1}{\lambda + 1} = \frac{-2\lambda + 1}{\lambda + 1} \]
\[ y_Q = \frac{\lambda \cdot 1 + 1 \cdot 2}{\lambda + 1} = \frac{\lambda + 2}{\lambda + 1} \]

### Step 4: Substitute \(Q\) into the Line Equation
Since point \(Q\) lies on the line \(3x + 4y = 7\), we substitute \(x_Q\) and \(y_Q\) into the line equation:
\[ 3\left(\frac{-2\lambda + 1}{\lambda + 1}\right) + 4\left(\frac{\lambda + 2}{\lambda + 1}\right) = 7 \]

### Step 5: Simplify the Equation
Multiply through by \((\lambda + 1)\) to eliminate the denominator:
\[ 3(-2\lambda + 1) + 4(\lambda + 2) = 7(\lambda + 1) \]
Expanding this gives:
\[ -6\lambda + 3 + 4\lambda + 8 = 7\lambda + 7 \]
Combine like terms:
\[ -2\lambda + 11 = 7\lambda + 7 \]
Rearranging gives:
\[ -2\lambda - 7\lambda = 7 - 11 \]
\[ -9\lambda = -4 \]
Thus,
\[ \lambda = \frac{4}{9} \]

### Step 6: Determine the Ratio
The ratio in which the line divides the segment is:
\[ \lambda : 1 = \frac{4}{9} : 1 \]
This can be expressed as:
\[ 4 : 9 \]

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