Question

Let the image of the point \( (1, 0, 7) \) in the line \[ \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \] 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 \( \frac{2\pi}{3} \) and \( \frac{3\pi}{4} \) with the y-axis and z-axis respectively and an acute angle with the x-axis?

• A. \( (1, -2, 1 + \sqrt{2}) \)
• B. \( (2, 1, 2 - \sqrt{2}) \)
• C. \( (3, 4, 3 - 2\sqrt{2}) \)
• D. \( (3, -4, 3 + 2\sqrt{2}) \)

Answer 1

The Correct Option is C

To find the image of the point \((1, 0, 7)\) in the line \(\frac{\vec{r}}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}\), let us proceed with a step-by-step approach.

Equation of the Line
The line \(L_1\) is given by:  
\(\frac{\vec{r}}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} = \lambda\)
with direction vector \(\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}\).

Finding the Foot of Perpendicular (Point \(M\))
Let \(M\) be the foot of the perpendicular from \(P(1, 0, 7)\) to \(L_1\) with coordinates  
\((1 + \lambda, 1 + 2\lambda, 2 + 3\lambda).\)

The vector \(\vec{PM}\) is:  
\(\vec{PM} = (\lambda - 1)\hat{i} + (1 + 2\lambda)\hat{j} + (3\lambda - 5)\hat{k}.\)

Condition of Perpendicularity
Since \(\vec{PM}\) is perpendicular to the direction vector \(\vec{b}\), we have:  
\(\vec{PM} \cdot \vec{b} = 0.\)
Expanding, we get:  
\((\lambda - 1) + 2(1 + 2\lambda) + 3(3\lambda - 5) = 0.\)
Simplifying, we find:  
\(14\lambda - 14 = 0 \implies \lambda = 1.\)

Thus, \(M = (2, 3, 5)\).

Finding the Image Point \(Q(\alpha, \beta, \gamma)\)
Since \(M\) is the midpoint of \(P\) and \(Q\), we have:  
\(Q = 2M - P = (1, 6, 3).\)

Therefore, \((\alpha, \beta, \gamma) = (1, 6, 3)\).

Verifying the Required Point on the Line
We need to find a point on the line passing through \((1, 6, 3)\) that makes angles \(\frac{\pi}{4}\) and \(\frac{\pi}{4}\) with the y-axis and z-axis, respectively, and an acute angle with the x-axis.  
After verifying, the point that satisfies these conditions is:  
\(\text{Option (3): } (3, 4, 3 - 2\sqrt{3}).\)

Thus, the correct answer is: \( (3, 4, 3 - 2\sqrt{2}) \)