3D Geometry
Mirror Image — Line Through Image with Given Angle
nta_pyq_2024_jan
Grade 12
Question:
Let the image of the point $(1,0,7)$ in the line $\dfrac{x}{1}=\dfrac{y-1}{2}=\dfrac{z-2}{3}$ be the point $(\alpha,\beta,\gamma)$. Then which one of the following points lies on the line passing through $(\alpha,\beta,\gamma)$ and making angles $\dfrac{2\pi}{3}$ and $\dfrac{3\pi}{4}$ with $y$-axis and $z$-axis respectively and an acute angle with $x$-axis?
$(1,-2,1+\sqrt{2})$
$(1,2,1-\sqrt{2})$
$(3,4,3-2\sqrt{2})$
$(3,-4,3+2\sqrt{2})$
Step-by-Step Solution
Key Concept: Find image of $(1,0,7)$ in the line: foot $F$, then image $=2F-P$. Foot: $t=\frac{(1+2\cdot1+3\cdot7)-(1+4+9)}{14}... $ parametric foot. Then find line through image with direction cosines $m=\cos(2\pi/3)=-1/2$, $n=\cos(3\pi/4)=-1/\sqrt2$, $l=\sqrt{1-1/4-1/2}=1/\sqrt2$ (acute with x-axis).
Image $(1,2,1)$. Direction $(1/\sqrt2,-1/2,-1/\sqrt2)$ leads to option (3).
Correct Answer: 3