3D Geometry
Mirror Image
MMTS_Full_Test_05
Grade 12
Question:
Let $(\alpha,\beta,\gamma)$ be the mirror image of $(1,-2,3)$ in the line $\dfrac{x+1}{2}=\dfrac{y-3}{4}=\dfrac{z+2}{3}$. Then $2\alpha+4\beta+3\gamma+2=$
Step-by-Step Solution
Key Concept: Find foot of perpendicular from point to line; then reflect
Foot: let parameter $t$: $P=(2t-1,4t+3,3t-2)$. $\overrightarrow{AP}\cdot\vec{d}=0$: $(2t-2)2+(4t+5)4+(3t-5)3=0$: $4t-4+16t+20+9t-15=0\Rightarrow 29t+1=0\Rightarrow t=-1/29$. Mirror point and $2\alpha+4\beta+3\gamma+2=11$.
Correct Answer: 2