Medium INTEGER +4 / -1 PYQ · JEE Mains 2023

$A(2,6,2), B(-4,0, \lambda), C(2,3,-1)$ 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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