The line `x-y+k=0` passes through the point which divides the segment joining the points (2,3) and (4,5) in the the ratio 2 : 3. Find the value of k.

Answer

To find the value of \( k \) in the equation of the line \( x - y + k = 0 \) that passes through the point dividing the segment joining the points \( (2, 3) \) and \( (4, 5) \) in the ratio \( 2:3 \), we can follow these steps:

### Step 1: Find the coordinates of the point that divides the segment in the ratio \( 2:3 \).

Given points:
- \( A(2, 3) \) and \( B(4, 5) \)

Let \( C(x, y) \) be the point that divides the segment \( AB \) in the ratio \( 2:3 \). The coordinates of point \( C \) can be calculated using the section formula:

\[ C\left( \frac{m_2 x_1 + m_1 x_2}{m_1 + m_2}, \frac{m_2 y_1 + m_1 y_2}{m_1 + m_2} \right) \]

where \( m_1 = 2 \), \( m_2 = 3 \), \( (x_1, y_1) = (2, 3) \), and \( (x_2, y_2) = (4, 5) \).

Calculating the x-coordinate of \( C \):

\[ x = \frac{3 \cdot 2 + 2 \cdot 4}{2 + 3} = \frac{6 + 8}{5} = \frac{14}{5} \]

Calculating the y-coordinate of \( C \):

\[ y = \frac{3 \cdot 3 + 2 \cdot 5}{2 + 3} = \frac{9 + 10}{5} = \frac{19}{5} \]

Thus, the coordinates of point \( C \) are \( \left( \frac{14}{5}, \frac{19}{5} \right) \).

### Step 2: Substitute the coordinates of point \( C \) into the line equation.

The line equation is given by:

\[ x - y + k = 0 \]

Substituting \( x = \frac{14}{5} \) and \( y = \frac{19}{5} \):

\[ \frac{14}{5} - \frac{19}{5} + k = 0 \]

### Step 3: Simplify the equation.

Calculating the left-hand side:

\[ \frac{14 - 19}{5} + k = 0 \]
\[ \frac{-5}{5} + k = 0 \]
\[ -1 + k = 0 \]

### Step 4: Solve for \( k \).

Rearranging gives:

\[ k = 1 \]

Thus, the value of \( k \) is \( 1 \).

### Final Answer:
The value of \( k \) is \( 1 \).

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