The ratio in which the line 2x + y - 4 = 0 divides the line segment joining the points A(2, -2) and B(3, 7) is
(a)`3 : 7`
(b)`4 : 7`
(c)`2 : 9`
(d)`4 : 9`

Options

A. `3 : 7`
B. `4 : 7`
C. `2 : 9`
D. `4 : 9`

Answer

To find the ratio in which the line \(2x + y - 4 = 0\) divides the line segment joining the points \(A(2, -2)\) and \(B(3, 7)\), we can follow these steps:

### Step 1: Identify the coordinates of points A and B
- Point A is given as \(A(2, -2)\).
- Point B is given as \(B(3, 7)\).

### Step 2: Assign the ratio
Let the ratio in which the line divides the segment \(AB\) be \(k : 1\).

### Step 3: Use the section formula
The coordinates of the point \(P\) that divides the line segment joining \(A\) and \(B\) in the ratio \(k : 1\) can be calculated using the section formula:
\[ P(x, y) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Where \(m = k\), \(n = 1\), \(x_1 = 2\), \(y_1 = -2\), \(x_2 = 3\), and \(y_2 = 7\).

### Step 4: Calculate the x-coordinate of P
Using the section formula for the x-coordinate:
\[ x = \frac{k \cdot 3 + 1 \cdot 2}{k + 1} = \frac{3k + 2}{k + 1} \]

### Step 5: Calculate the y-coordinate of P
Using the section formula for the y-coordinate:
\[ y = \frac{k \cdot 7 + 1 \cdot (-2)}{k + 1} = \frac{7k - 2}{k + 1} \]

### Step 6: Substitute the coordinates into the line equation
Now, substitute \(x\) and \(y\) into the line equation \(2x + y - 4 = 0\):
\[ 2\left(\frac{3k + 2}{k + 1}\right) + \left(\frac{7k - 2}{k + 1}\right) - 4 = 0 \]

### Step 7: Simplify the equation
Multiply through by \(k + 1\) to eliminate the denominator:
\[ 2(3k + 2) + (7k - 2) - 4(k + 1) = 0 \]
This simplifies to:
\[ 6k + 4 + 7k - 2 - 4k - 4 = 0 \]
Combine like terms:
\[ (6k + 7k - 4k) + (4 - 2 - 4) = 0 \]
\[ 9k - 2 = 0 \]

### Step 8: Solve for k
Solving for \(k\):
\[ 9k = 2 \implies k = \frac{2}{9} \]

### Step 9: Determine the ratio
The ratio in which the line divides the segment \(AB\) is \(k : 1\) or \(\frac{2}{9} : 1\). This can be expressed as:
\[ 2 : 9 \]

### Conclusion
Thus, the ratio in which the line \(2x + y - 4 = 0\) divides the line segment joining the points \(A(2, -2)\) and \(B(3, 7)\) is \(2 : 9\).

### Final Answer
The correct option is (c) \(2 : 9\).

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