Ratio in which the join of (2, 1) and `(-1, 2)` is divided by the line `x + 3y + 5 = 0` is

Options

A. `1 : 2` internally
B. `1 : 3` externally
C. `2 : 1` externally
D. None of these

Answer

To find the ratio in which the line segment joining the points (2, 1) and (-1, 2) is divided by the line \(x + 3y + 5 = 0\), we can follow these steps:

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

### Step 2: Use the section formula
The section formula states that if a point \(P\) divides the line segment joining points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m:n\), then the coordinates of point \(P\) are given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

### Step 3: Set up the coordinates for point P
Using the coordinates of points \(A\) and \(B\):
- \(x_1 = 2\), \(y_1 = 1\)
- \(x_2 = -1\), \(y_2 = 2\)

The coordinates of point \(P\) will be:
\[ P\left(\frac{m(-1) + n(2)}{m+n}, \frac{m(2) + n(1)}{m+n}\right) \]

### Step 4: Substitute into the line equation
Since point \(P\) lies on the line \(x + 3y + 5 = 0\), we substitute the coordinates of \(P\) into the line equation:
\[ \frac{m(-1) + n(2)}{m+n} + 3\left(\frac{m(2) + n(1)}{m+n}\right) + 5 = 0 \]

### Step 5: Simplify the equation
Multiply through by \(m+n\) to eliminate the denominator:
\[ m(-1) + n(2) + 3(m(2) + n(1)) + 5(m+n) = 0 \]
Expanding this gives:
\[ -m + 2n + 6m + 3n + 5m + 5n = 0 \]
Combine like terms:
\[ (10m + 10n) = 0 \]

### Step 6: Solve for the ratio
This simplifies to:
\[ 10m + 10n = 0 \implies m + n = 0 \implies m = -n \]
Thus, the ratio \( \frac{m}{n} = -1 \).

### Step 7: Interpret the ratio
Since \(m = -n\), we can express this as:
\[ \frac{m}{n} = -1 \implies m:n = 1:1 \]
This indicates that the line divides the segment externally in the ratio \(1:1\).

### Final Answer
The ratio in which the line divides the segment joining the points (2, 1) and (-1, 2) is \(1:1\).

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