Vectors & 3D Geometry
Perpendicular bisector plane and volume of tetrahedron
MJAT_TS3_P1
Grade 12
Question:
Let $P$ be the plane consisting of all points equidistant from $A(-4,2,1)$ and $B(2,-4,3)$. Let $Q$ denote the plane $x-y+Cz=1$, where $C\in\mathbb{R}$.
A) If $C=\dfrac{1}{3}$, then plane $P$ is parallel to $Q$
B) If $C=-1$, then plane $P$ is perpendicular to $Q$
C) Volume of tetrahedron formed by plane $P$ with coordinate planes is $\dfrac{28}{1}$
D) If line $L:\dfrac{x-1}{1}=\dfrac{y+3}{2}=\dfrac{z-7}{-1}$ intersects plane $P$ at $R(\alpha,\beta,\gamma)$, then $\alpha+\beta+\gamma=12$
Step-by-Step Solution
Key Concept: Midpoint of $AB$: $(-1,-1,2)$. Direction $\overrightarrow{AB}=(6,-6,2)\propto(3,-3,1)$. Plane $P$: $3(x+1)-3(y+1)+(z-2)=0\Rightarrow 3x-3y+z=2$.
A ✓, D ✓. Answer: A, D.
Correct Answer: AD