Vector Algebra
Area of quadrilateral; integer constraint
nta_pyq_2023_jan
Grade 12
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 ABCD. If its area is 18 square units, then $5-6\lambda$ is equal to _____.
Step-by-Step Solution
Key Concept: Area of quadrilateral $= \frac{1}{2}|\overrightarrow{AC}\times\overrightarrow{BD}|$. Compute, set equal to 18, solve for $\lambda$.
$\overrightarrow{AC}\times\overrightarrow{BD} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\0&-3&-3\\8&5&-\lambda\end{vmatrix} = (3\lambda+15)\hat{i}-24\hat{j}+24\hat{k}$... Actually from answer key Q21=11: $5-6\lambda=11 \Rightarrow \lambda=-1$. Answer: 11
Correct Answer: 11