MCQ
$A (2,6,2), B (-4,0, \lambda), C (2,3,-1)$ અને $D (4,5,0)$, $|\lambda| \leq 5$ એ ચતુષ્કોણ $A B C D$ ના શિરોબિંદુઓ છે. જો તેનું ક્ષેત્રફળ $18$ ચોરસ એકમ હોય તો $5-6 \lambda$ ની કિમંત મેળવો.
- A$10$
- B$12$
- ✓$11$
- D$14$
$\text { Area }=\frac{1}{2}|\overrightarrow{B D} \times \overline{A C}|=18$
$\overrightarrow{A C} \times \overline{B D}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 0 & -3 & -3 \\ 8 & 5 & -\lambda \end{array}\right|$
$=(3 \lambda+15) \hat{i}-\hat{j}(-24)+\hat{k}(-24)$
$\overline{A C} \times \overline{B D}=(3 \lambda+15) \hat{i}+24 \hat{j}-24 \hat{k}$
$=\sqrt{(3 \lambda+15)^2+(24)^2+(24)^2}=36 $
$ =\lambda^2+10 \lambda+9=0 $
$=\lambda=-1,-9$
$|\lambda| \leq 5 \Rightarrow \lambda=-1$
$5-6 \lambda=5-6(-1)=11$
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