Let $\vec{OA}=\vec a,\,\vec{OB}=10\vec a+2\vec b$ and $\vec{OC}=\vec b$ where, O is the origin. Let p denote the area of the quadrilateral OABC, and let q denote the area of the parallelogram with OA and OC as adjacent sides. If p = kq, then k is:

6

Area of Δ(OAB) = $\frac{1}{2}|\vec a×(10\vec a+2\vec b)|=|\vec a×\vec b|$

Area of Δ(OBC) = $\frac{1}{2}|(10\vec a+2\vec b)×\vec b|=5|\vec a×\vec b|$

⇒ Area of parallelogram $\vec a$ and $\vec b$ as sides = $|\vec a×\vec b|=q$

⇒ p = 6q. Hence k = 6