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 :

Straight Lines and Pair of Straight Lines PYQ — JEE Main 2012 · Shift 0 · Q39 with Solution

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Correct answer: C

Let the points be $A(1,1),\quad B(2,4).$
We need the point that divides the line segment joining $A$ and $B$ in the ratio $3:2$ internally.

Using the section formula, if $P$ divides $AB$ in the ratio $3:2$ , then $P\left(\frac{3x_2+2x_1}{3+2},\;\frac{3y_2+2y_1}{3+2}\right).$

Substituting $A(1,1)$ and $B(2,4)$ :
$=\left(\frac{6+2}{5},\;\frac{12+2}{5}\right) =\left(\frac{8}{5},\;\frac{14}{5}\right).$$$
Since the line $2x+y=k$ passes through this point, substitute $x=\frac{8}{5}$ and $y=\frac{14}{5}$ :
$=\frac{16}{5}+\frac{14}{5} =\frac{30}{5}=6.$$$
Therefore, $k=6.$

So the correct option is C.