Find the ration in which the line joining the points `A(1,2)` and`B(-3,4)` is divided by the line.
`x+y-5=0`

Answer

To find the ratio in which the line joining the points A(1, 2) and B(-3, 4) is divided by the line \(x + y - 5 = 0\), we can follow these steps:

### Step 1: Define the Points and the Line
Let the points be:
- \( A(1, 2) \)
- \( B(-3, 4) \)

The equation of the line is:
\[ x + y - 5 = 0 \]

### Step 2: Assume the Division Ratio
Assume the line divides the segment AB in the ratio \( \lambda : 1 \). Let the point of division be \( P(h, k) \).

### Step 3: Use the Section Formula
According to the section formula, the coordinates of point \( P \) dividing the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \) are given by:
\[ h = \frac{mx_2 + nx_1}{m+n} \]
\[ k = \frac{my_2 + ny_1}{m+n} \]
For our case, \( m = \lambda \) and \( n = 1 \):
\[ h = \frac{\lambda(-3) + 1(1)}{\lambda + 1} = \frac{-3\lambda + 1}{\lambda + 1} \]
\[ k = \frac{\lambda(4) + 1(2)}{\lambda + 1} = \frac{4\lambda + 2}{\lambda + 1} \]

### Step 4: Substitute into the Line Equation
Since point \( P(h, k) \) lies on the line \( x + y - 5 = 0 \), we substitute \( h \) and \( k \) into the line equation:
\[ h + k - 5 = 0 \]
Substituting the expressions for \( h \) and \( k \):
\[ \frac{-3\lambda + 1}{\lambda + 1} + \frac{4\lambda + 2}{\lambda + 1} - 5 = 0 \]

### Step 5: Combine the Fractions
Combine the fractions:
\[ \frac{-3\lambda + 1 + 4\lambda + 2 - 5(\lambda + 1)}{\lambda + 1} = 0 \]
This simplifies to:
\[ \frac{-3\lambda + 1 + 4\lambda + 2 - 5\lambda - 5}{\lambda + 1} = 0 \]
\[ \frac{-4\lambda - 2}{\lambda + 1} = 0 \]

### Step 6: Solve for \( \lambda \)
Setting the numerator to zero:
\[ -4\lambda - 2 = 0 \]
\[ -4\lambda = 2 \]
\[ \lambda = -\frac{1}{2} \]

### Step 7: Determine the Ratio
Since \( \lambda = -\frac{1}{2} \), the ratio in which the line divides the segment AB is:
\[ \text{Ratio} = \left| \frac{\lambda}{1} \right| = \left| -\frac{1}{2} \right| : 1 = 1 : 2 \]
Since \( \lambda \) is negative, the division is external.

### Final Answer
The line divides the segment joining points A and B in the ratio \( 1 : 2 \) externally.

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