$A B C D$ is a quadrilateral with $\to A B = \to a, \to A D = \to b, ¯ ¯¯¯¯¯¯ ¯ A C = 2 \to a + 3 \to b$ . 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

5

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{5}{2}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{1}{2}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $\frac{5}{2}$
$- - \to B D = - - \to A D - - - \to A B$
$= \to b - \to a$
Area of parallelogram $= \frac{1}{2} ∣ ∣ ∣ - - \to B D \times - - \to A C ∣ ∣ ∣$
$= \frac{1}{2} ∣ ∣ (\to b - \to a) \times (2 \to a + 3 \to b) ∣ ∣$
$= \frac{1}{2} ∣ ∣ 2 \to b \times \to a - 3 \to a \times \to b ∣ ∣$
$= \frac{5}{2} ∣ ∣ \to b \times \to a ∣ ∣$
Area of parallelogram $= ∣ ∣ ∣ - - \to A B \times - - \to A D ∣ ∣ ∣$
$= ∣ ∣ a \times \to b ∣ ∣$
$P = \frac{\frac{5}{2} ∣ ∣ \to b \times \to a ∣ ∣}{∣ ∣ \to a \times \to b ∣ ∣} = \frac{5}{2}$

Suggest Corrections

Similar questions

A B C D

- - \to A B = \to a, - - \to A D = \to b, - - \to A C = 2 \to a + 3 \to b

p

A B

A D

p

	List 1		List 2
A.	The volume of the parallelopiped determined by vectors $\to a, \to b$ and $\to c$ is 2. Then, the volume of the parallelopiped determined by vectors $2 (\to a \times \to b), 3 (\to b \times \to c)$ and $(\to c \times \to a)$ is	1.	$100$
B.	The volume of the parallelopiped determined by vectors $\to a, \to b$ and $\to c$ is 5. Then, the volume of the parallelopiped determined by vectors $3 (\to a + \to b), (\to b + \to c)$ and $2 (\to c + \to a)$ is	2.	$30$
C.	The area of a triangle with adjacent sides determined by vectors $\to a$ and $\to b$ is 20. Then, the area of the triangle with adjacent sides determined by vectors $(2 \to a + 3 \to b)$ and $(\to a - \to b)$ is	3.	$24$
D.	The area of a parallelogram with adjacent sides determined by vectors $\to a$ and $\to b$ is $30$ . Then, the area of the parallelogram with adjacent sides determined by vectors $(\to a + \to b)$ and $\to a$ is	4.	$60$