Key Concepts and Formulas

• Section Formula (Internal Division): If point $P(x, y)$ divides the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in the ratio $m : n$, then $P(x, y) = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)$.
• Point on a Line: If a point $(x_0, y_0)$ lies on the line $ax + by + c = 0$, then $ax_0 + by_0 + c = 0$.

Step-by-Step Solution

Step 1: Identify the Given Information

We are given:

• Two points: $A(1, 1)$ and $B(2, 4)$.
• The ratio: $m : n = 3 : 2$.
• The equation of the line: $2x + y = k$.
• We need to find the value of $k$.

Step 2: Find the Coordinates of the Dividing Point using the Section Formula

We need to find the coordinates of the point $P(x, y)$ that divides the line segment joining $A(1, 1)$ and $B(2, 4)$ in the ratio $3 : 2$.

• Why this step? The problem states that the line $2x + y = k$ passes through this dividing point. Therefore, we need to find its coordinates.

Using the Section Formula: $x = \frac{m x_2 + n x_1}{m+n} = \frac{3(2) + 2(1)}{3+2} = \frac{6 + 2}{5} = \frac{8}{5}$ $y = \frac{m y_2 + n y_1}{m+n} = \frac{3(4) + 2(1)}{3+2} = \frac{12 + 2}{5} = \frac{14}{5}$

Therefore, the coordinates of the point $P$ are $\left(\frac{8}{5}, \frac{14}{5}\right)$.

Step 3: Substitute the Coordinates into the Line Equation

The line $2x + y = k$ passes through the point $P\left(\frac{8}{5}, \frac{14}{5}\right)$.

• Why this step? If a point lies on a line, its coordinates must satisfy the equation of the line. Substituting the coordinates of $P$ into the line equation will allow us to solve for $k$.

Substitute $x = \frac{8}{5}$ and $y = \frac{14}{5}$ into $2x + y = k$: $2\left(\frac{8}{5}\right) + \frac{14}{5} = k$ $\frac{16}{5} + \frac{14}{5} = k$ $\frac{30}{5} = k$ $k = 6$

Common Mistakes & Tips

• Section Formula Order: Ensure correct substitution into the section formula. $m$ is associated with $(x_2, y_2)$ and $n$ with $(x_1, y_1)$.
• Arithmetic Errors: Double-check fraction arithmetic to avoid mistakes.

Summary

We used the section formula to find the coordinates of the point dividing the line segment in the given ratio. Then, we substituted these coordinates into the equation of the line to solve for the unknown constant $k$. The value of $k$ is 6.

The final answer is $\boxed{6}$, which corresponds to option (C).