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

The required point is the internal division point of $$(1,1)$$ and $$(2,4)$$ in the ratio $$3:2$$ (first point’s weight 3, second point’s weight 2).

Section-formula for internal division:
If $$(x_1,y_1)$$ and $$(x_2,y_2)$$ are divided in ratio $$m:n$$, the coordinates are
$$\left(\frac{mx_2+nx_1}{m+n},\; \frac{my_2+ny_1}{m+n}\right).$$

Here $$m=3,\; n=2,\;(x_1,y_1)=(1,1),\;(x_2,y_2)=(2,4).$$
Hence the coordinates become
$$x=\frac{3\cdot2+2\cdot1}{3+2}= \frac{6+2}{5}= \frac{8}{5},$$
$$y=\frac{3\cdot4+2\cdot1}{3+2}= \frac{12+2}{5}= \frac{14}{5}.$$

This point $$(\frac{8}{5},\frac{14}{5})$$ lies on the line $$2x+y=k$$, so substitute:

$$2\left(\frac{8}{5}\right)+\frac{14}{5}=k \; \Longrightarrow \; \frac{16}{5}+\frac{14}{5}=k \; \Longrightarrow \; \frac{30}{5}=k.$$

Thus $$k=6$$.

Option C which is: $$6$$