In what ratio does the line `2x+y-4` divides the join of `P(2,-2)andQ(3,7)` ?

Answer

To find the ratio in which the line \(2x + y - 4 = 0\) divides the line segment joining the points \(P(2, -2)\) and \(Q(3, 7)\), we can use the section formula. Here’s a step-by-step solution:

### Step 1: Understand the Section Formula
The section formula states that if a point \(R\) divides the line segment joining points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) in the ratio \(m:n\), then the coordinates of point \(R\) can be calculated as:
\[ R\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

### Step 2: Assign Coordinates
Let \(P(2, -2)\) be \((x_1, y_1)\) and \(Q(3, 7)\) be \((x_2, y_2)\). We will assume that point \(R\) divides \(PQ\) in the ratio \(k:1\).

### Step 3: Apply the Section Formula
Using the section formula, the coordinates of point \(R\) can be expressed as:
\[ R\left(\frac{3k + 2}{k + 1}, \frac{7k - 2}{k + 1}\right) \]

### Step 4: Substitute into the Line Equation
Since point \(R\) lies on the line \(2x + y - 4 = 0\), we substitute the coordinates of \(R\) into this equation:
\[ 2\left(\frac{3k + 2}{k + 1}\right) + \left(\frac{7k - 2}{k + 1}\right) - 4 = 0 \]

### Step 5: Simplify the Equation
Multiply through by \(k + 1\) to eliminate the denominator:
\[ 2(3k + 2) + (7k - 2) - 4(k + 1) = 0 \]
Expanding this gives:
\[ 6k + 4 + 7k - 2 - 4k - 4 = 0 \]
Combine like terms:
\[ (6k + 7k - 4k) + (4 - 2 - 4) = 0 \]
This simplifies to:
\[ 9k - 2 = 0 \]

### Step 6: Solve for \(k\)
Now, solve for \(k\):
\[ 9k = 2 \implies k = \frac{2}{9} \]

### Step 7: Determine the Ratio
The ratio in which the line divides the segment \(PQ\) is \(k:1\), which is:
\[ \frac{2}{9}:1 \text{ or } 2:9 \]

Thus, the line \(2x + y - 4 = 0\) divides the line segment joining \(P(2, -2)\) and \(Q(3, 7)\) in the ratio \(2:9\).

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