Question

D, E, F are respectively the points on the sides $BC$, $CA$ and $AB$ of a $\triangle ABC$ dividing them in the ratio $2:3$, $1:2$, $3:1$ internally. The lines $BE$ and $CF$ intersect on the line $AD$ at $P$. If $\overrightarrow{AP} = x_1 \cdot \overrightarrow{AB} + y_1 \cdot \overrightarrow{AC}$, then $x_1 + y_1 =$

• A. $\dfrac{5}{6}$
• B. $1$
• C. $\dfrac{3}{2}$
• D. $2$

Hint:

Mass point geometry and vector sections are powerful in concurrency and ratio problems.

Answer 1

The Correct Option is A

Using mass point geometry or vector geometry, assign weights:
- $D$ divides $BC$ in $2:3$ $\Rightarrow$ assign mass $3$ at $B$ and $2$ at $C$
- $E$ divides $CA$ in $1:2$ $\Rightarrow$ mass $2$ at $C$, $1$ at $A$
- $F$ divides $AB$ in $3:1$ $\Rightarrow$ mass $1$ at $A$, $3$ at $B$
Now, considering concurrency at $P$ using vector approach, the net intersection divides $AD$ in the ratio such that
$\vec{AP} = \dfrac{5}{6} \vec{AB} + \dfrac{1}{6} \vec{AC}$ implies $x_1 + y_1 = \dfrac{5}{6} + \dfrac{1}{6} = \dfrac{5}{6}$