A(2, 6, 2), B(- 4, 0, ), C(2, 3, - 1) and D(4, 5, 0), 5 are the vertices of a quadrilateral ABCD. If its area is 18 square units, then | 5 - 6 | is equal to ______ . [2023]

Question

A(2, 6, 2), B(- 4, 0, $λ$ ), C(2, 3, - 1) and D(4, 5, 0), $| λ |$ $\leq 5$ are the vertices of a quadrilateral ABCD. If its area is 18 square units, then $| 5 - 6 λ |$ is equal to ______ . [2023]

Smart Achievers · Accepted Answer

(11)

$Area of quadrilateral = \frac{1}{2}$ $| \vec{A C} \times \vec{B D} |$

$\Rightarrow 18 = \frac{1}{2}$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & - 3 & - 3 \\ 8 & 5 & - λ \end{matrix} |$

$\Rightarrow 36 =$ $| (3 λ + 15) \hat{i} - \hat{j} (0 + 24) + \hat{k} (0 + 24) |$

$\Rightarrow 36 = \sqrt{{(3 λ + 15)}^{2} + {(- 24)}^{2} + 24^{2}}$

$\Rightarrow λ^{2} + 10 λ + 9 = 0$

$\Rightarrow λ = - 1, - 9 \Rightarrow$ $| λ |$ $\leq 5 \Rightarrow λ = - 1$

$\Rightarrow 5 - 6 λ = 5 - 6 (- 1) = 11$

Solution

Q.
A(2, 6, 2), B(- 4, 0, $λ$ ), C(2, 3, - 1) and D(4, 5, 0), $| λ |$ $\leq 5$ are the vertices of a quadrilateral ABCD. If its area is 18 square units, then $| 5 - 6 λ |$ is equal to ______ . [2023]

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Solution

Q. A(2, 6, 2), B(- 4, 0, λ), C(2, 3, - 1) and D(4, 5, 0), |λ|≤5 are the vertices of a quadrilateral ABCD. If its area is 18 square units, then |5-6λ| is equal to ______ . [2023]

Ans. (11) Area of quadrilateral=12|AC→×BD→| ⇒18=12|i^j^k^0-3-385-λ| ⇒36=|(3λ+15)i^-j^(0+24)+k^(0+24)| ⇒36=(3λ+15)2+(-24)2+242 ⇒λ2+10λ+9=0 ⇒λ=-1, -9⇒|λ|≤5⇒λ=-1 ⇒ 5-6λ=5-6(-1)=11

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Q.
A(2, 6, 2), B(- 4, 0, $λ$ ), C(2, 3, - 1) and D(4, 5, 0), $| λ |$ $\leq 5$ are the vertices of a quadrilateral ABCD. If its area is 18 square units, then $| 5 - 6 λ |$ is equal to ______ . [2023]