JEE 2023 — Mathematics PYQ

Maana ki vertices $A,B,C,$ aur $D$ ke position vectors kramashah $\vec{a},\vec{b},\vec{c},$ aur $\vec{d}$ hain.

Step 1: Midpoints ke position vectors nikalna

Kyunki $E$, $AC$ ka midpoint hai aur $F$, $BD$ ka midpoint hai:

Step 2: Di gayi equation ke vectors ko solve karna

LHS ko position vectors ke roop mein likhte hain:

Ab terms ko regroup karte hain:

Step 3: Midpoint vectors se compare karna

Upar wale values se hum jante hain ki $(\vec{b}+\vec{d})=2\vec{f}$ aur $(\vec{a}+\vec{c})=2\vec{e}$. Inhe equation mein rakhne par:

Step 4: $k$ ki value nikalna

Hume diya gaya hai:

Kyunki $\overrightarrow{FE}=-\overrightarrow{EF}$, toh:

Isliye, $k=-4$.

Correct Option: (4) $-4$