Question

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

The correct answer is 11.
A(2,6,2)B(−4,0,λ),C(2,3,−1)D(4,5,0)
Area =21​∣BD×AC∣=18
AC×BD=∣∣​i^08​j^​−35​k^−3−λ​∣∣​
=(3λ+15)i^−j^​(−24)+k^(−24)
AC×BD=(3λ+15)i^+24j^​−24k^
=(3λ+15)2+(24)2+(24)2​=36
=λ2+10λ+9=0
=λ=−1,−9
∣λ∣≤5⇒λ=−1
5−6λ=5−6(−1)=11