Question

If the line $2x+y=k$ passes through the point which divides the segment joining the points $\left(1,1\right)$ and $\left(2,4\right)$ in the ratio $3:2$, then $k$ is equal to

• A. $\frac{29}{5}$
• B. $5$
• C. $6$
• D. $\frac{11}{15}$

Answer 1

The correct option is C

$6$

The explanation for the correct option:

The given points are $\left(1,1\right)$ and $\left(2,4\right)$.

The point which divides the segment joining the points $\left(1,1\right)$ and $\left(2,4\right)$ in the ratio $3:2$ can be given by $P\equiv \left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right)$.

$$\begin{gathered} \Rightarrow P\equiv \left(\frac{\left(3\times 2\right)+\left(2\times 1\right)}{3+2},\frac{\left(3\times 4\right)+\left(2\times 1\right)}{3+2}\right) \\ \Rightarrow P\equiv \left(\frac{8}{5},\frac{14}{5}\right) \end{gathered}$$

The point $P$ lies on the straight line $2x+y=k$.

Thus, $2\left(\frac{8}{5}\right)+\left(\frac{14}{5}\right)=k$.

$$\begin{gathered} \Rightarrow \frac{16}{5}+\frac{14}{5}=k \\ \Rightarrow k=\frac{16+14}{5} \\ \Rightarrow k=\frac{30}{5} \\ \Rightarrow k=6 \end{gathered}$$

Therefore, the value of $k$ is $6$.

Hence, option (C) $6$ is the correct answer.