JEE MAIN - Mathematics (2024 - 29th January Evening Shift - No. 14)
Let $$\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=12 \vec{a}+4 \vec{b} \text { and } \overrightarrow{O C}=\vec{b}$$, where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then $$\mathrm{{{area\,of\,the\,quadrilateral\,OA\,BC} \over {area\,of\,S}}}$$ is equal to _________.
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Explanation
_29th_January_Evening_Shift_en_14_1.png)
Area of parallelogram, $$S=|\vec{a} \times \vec{b}|$$
Area of quadrilateral $$=\operatorname{Area}(\triangle \mathrm{OAB})+\operatorname{Area}(\triangle \mathrm{OBC})$$
$$\begin{aligned} & =\frac{1}{2}\{|\vec{a} \times(12 \vec{a}+4 \vec{b})|+|\vec{b} \times(12 \vec{a}+4 \vec{b})|\} \\ & =8|(\vec{a} \times \vec{b})| \end{aligned}$$
$$\text { Ratio }=\frac{8|(\vec{a} \times \vec{b})|}{|(\vec{a} \times \vec{b})|}=8$$
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