$A(2,6,2), B(-4,0, \lambda), C(2,3,-1)$ and $D(4,5,0),|\lambda| \leq 5$ are the vertices of a quadrilateral $A B C D$. If its area is 18 square units, then $5-6 \lambda$ is equal to __________.
Answer (integer)
11
Solution
$$
\begin{aligned}
& \mathrm{A}(2,6,2) \quad \mathrm{B}(-4,0, \lambda), \mathrm{C}(2,3,-1) \mathrm{D}(4,5,0) \\\\
& \text { Area }=\frac{1}{2}|\overrightarrow{B D} \times \overrightarrow{A C}|=18 \\\\
& \overrightarrow{A C} \times \overrightarrow{B D}=\left|\begin{array}{ccc}
\hat{i} & j & k \\\\
0 & -3 & -3 \\\\
8 & 5 & -\lambda
\end{array}\right|
\end{aligned}
$$
<br/><br/>$$
\begin{aligned}
& =(3 \lambda+15) \hat{i}-\hat{j}(-24)+\hat{k}(-24) \\\\
& \overrightarrow{A C} \times \overrightarrow{B D}=(3 \lambda+15) \hat{i}+24 \hat{j}-24 \hat{k} \\\\
& =\sqrt{(3 \lambda+15)^2+(24)^2+(24)^2}=36 \\\\
& =\lambda^2+10 \lambda+9=0 \\\\
& =\lambda=-1,-9 \\\\
& |\lambda| \leq 5 \Rightarrow \lambda=-1 \\\\
& 5-6 \lambda=5-6(-1)=11
\end{aligned}
$$
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
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