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 \( \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \), we need to determine a point on the line that reflects this point across the line. Let's break down the problem step-by-step: 

1. The given line has a direction vector \((1, 2, 3)\) and passes through the point \((0, 1, 2)\).
2. Let \( (\alpha, \beta, \gamma) \) be the image of the point \((1, 0, 7)\) on the line.
3. The key property for the image is that the midpoint of \((1, 0, 7)\) and \((\alpha, \beta, \gamma)\) lies on the line.
4. We can substitute the parametric form of the line into the midpoint equation and solve for the parameter.
5. The parametric equation of the line is \( (x, y, z) = (t, 1 + 2t, 2 + 3t) \). The perpendicular from a point \((1, 0, 7)\) to the line will be closest at this point.
6. Finding the T closest to the original point invovles using the dot product of the direction with the vector from point to line. Setting that dot product to zero will give you the t value closest perpendicularly.

After finding the value of \(t\), we determine:

• \( x = t \)
• \( y = 1 + 2t \)
• \( z = 2 + 3t \)

These coordinates give us the point on the line. After performing calculations we find:

The coordinates of \((\alpha, \beta, \gamma)\) meet the criteria to satisfy both conditions of being midpoints. They are reflected appropriately.

Now, let's calculate the direction vector of the new line making angles \(\frac{2\pi}{3}\) with the y-axis and \(\frac{3\pi}{4}\) with the z-axis:

1. If the direction vector is \((a, b, c)\), then:
2. \(\cos \theta_y = \frac{b}{\sqrt{a^2 + b^2 + c^2}} = \cos \frac{2\pi}{3} = -\frac{1}{2}\)
3. \(\cos \theta_z = \frac{c}{\sqrt{a^2 + b^2 + c^2}} = \cos \frac{3\pi}{4} = -\frac{1}{\sqrt{2}}\)

We solve for \(a, b, c\) considering they form an acute angle with the x-axis:

1. Since the cosine of angles should sum to 1, we validate correctly that only one option meets this.

Given the constraints and angle conditions, we solve and find that:

The point that lies correctly is: \( (3, 4, 3 - 2\sqrt{2}) \).

Answer 2

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: \( (3, 4, 3 - 2\sqrt{2}) \)