$A B C D$ is a quadrilateral with $\bar{A B} = a, \bar{A D} = b, \bar{A C} = 2 a + 3 b$ If the area of parallelogram $A B C D$ is p times the area of the parallelogram with $A B, A D$ as adjacent sides, then p is equal to

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$5$

$\frac{5}{2}$

$1$

$\frac{1}{2}$

answer is B.

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Detailed Solution

$\frac{1}{2} | A C \times B D | = p | A B \times A D |$

$\Rightarrow \frac{1}{2} | (2 \bar{a} + 3 \bar{b}) \times (\bar{b} - \bar{a}) | = p | \bar{a} \times \bar{b} |$

$\Rightarrow \frac{1}{2} | 5 (\bar{a} \times \bar{b}) | = p | \bar{a} \times \bar{b} |$

$\Rightarrow \frac{1}{2} | \bar{a} \times \bar{b} | = p ∣ \bar{a} \times \bar{b} \Leftrightarrow p = \frac{5}{2}$

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