Question

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

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

Answer 1

The correct option is C $6$

Let $A\equiv (1,1)$ and $B\equiv (2,4)$

Let point $C$ divides $A$ and $B$ in the ratio $3:2$ internally, then
$C\equiv (\frac{3\cdot 2+2\cdot 1}{3+2},\frac{3\cdot 4+2\cdot 1}{3+2})$
$\Rightarrow C\equiv (\frac{8}{5},\frac{14}{5})$

Let point $C$ divides $A$ and $B$ in the ratio $3:2$ externally, then
$C\equiv (\frac{3\cdot 2-2\cdot 1}{3-2},\frac{3\cdot 4-2\cdot 1}{3-2})$
$\Rightarrow C\equiv (4,10)$

As the point $C$ lies on the line $2x+y=k,$ then
$2\times \frac{8}{5}+\frac{14}{5}=k$
$\Rightarrow k=\frac{30}{5}\Rightarrow k=6$

or $2\times 4+10=k$
$\Rightarrow k=18$