3D Geometry
Foot of Perpendicular — Triangle Area Constraint
nta_pyq_2023_apr
Grade 12
Question:
Foot of $\perp$ from $A(4,3,1)$ on $x-y+2z+3=0$ is $N$. $B(5,\alpha,\beta)$ on plane, area of $\triangle ABN=3\sqrt{2}$. Then $\alpha^2+\beta^2+\alpha\beta$ is equal to _____________
Step-by-Step Solution
Key Concept: Find $N$ via foot formula. $|BN|^2=18\cdot2/(\frac{1}{2}\cdot|BN|)$... Use $\text{Area}=\frac{1}{2}|AB||AN|\sin\theta$.
$7$.
Correct Answer: 7