KCET · Maths · Three Dimensional Geometry
The image of the point \( (1,6,3) \) in the line \( \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} \) is
- A \( (1,0,7) \)
- B \( (7,0,1) \)
- C \( (2,7,0) \)
- D \((-1,-6,-3) \)
Answer & Solution
Correct Answer
(A) \( (1,0,7) \)
Step-by-step Solution
Detailed explanation
Given line
\[
\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} \rightarrow(1)
\]
Point \( P(1,6,3) \) and reflection \( Q \).
Let \( \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}=\lambda \)
and point \( R \) on line. So, coordinates of \( R \) is
\( (\lambda, 1+2 \lambda, 2+3 \lambda) \)
Direction ratios of line is \( (1,2,3) \).
Since PR is perpendicular to the given line. Then, direction ratios of
\( \overline{P R}=(\lambda-1,2 \lambda-5,3 \lambda-1) \)
So, \( \lambda-1+2(2 \lambda-5)+3(3 \lambda-1)=0 \)
\( \Rightarrow 14 \lambda=14 \Rightarrow \lambda=1 \)
Therefore, \( R(1,3,5) \) and \( Q(1,0,7) \)
\[
\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} \rightarrow(1)
\]
Point \( P(1,6,3) \) and reflection \( Q \).
Let \( \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}=\lambda \)
and point \( R \) on line. So, coordinates of \( R \) is
\( (\lambda, 1+2 \lambda, 2+3 \lambda) \)
Direction ratios of line is \( (1,2,3) \).
Since PR is perpendicular to the given line. Then, direction ratios of
\( \overline{P R}=(\lambda-1,2 \lambda-5,3 \lambda-1) \)
So, \( \lambda-1+2(2 \lambda-5)+3(3 \lambda-1)=0 \)
\( \Rightarrow 14 \lambda=14 \Rightarrow \lambda=1 \)
Therefore, \( R(1,3,5) \) and \( Q(1,0,7) \)
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