Question

Let the line \( L \) intersect the lines
\( x - 2 = -y = z - 1, \quad 2 (x + 1) = 2(y - 1) = z + 1 \)
and be parallel to the line
\[ \frac{x-2}{3} = \frac{y-1}{1} = \frac{z-2}{2}. \]
Then which of the following points lies on \( L \)?

• A. \(\left( \frac{1}{3}, 1, 1 \right)\)
• B. \(\left( \frac{1}{3}, 1, -1 \right)\)
• C. \(\left( \frac{1}{3}, -1, -1 \right)\)
• D. \(\left( \frac{1}{3}, -1, 1 \right)\)

Answer 1

The Correct Option is B

The given line \( L \) is parallel to the line:

\[ \frac{x - 2}{3} = \frac{y - 1}{1} = \frac{z - 2}{2}, \]

which implies the direction ratios of \( L \) are \( 3 : 1 : 2 \).

Assume that \( L \) intersects the first line \( x - 2 = -y = z - 1 \) at some point \( P \). From the equation of the line:

\[ x - 2 = -y = z - 1 = k. \]

This gives:

\[ x = 2 + 3k, \quad y = -k, \quad z = 1 + k. \]

Next, assume \( L \) also intersects the second line \( 2(x + 1) = 2(y - 1) = z + 1 \) at some point \( Q \). From the equation of the line:

\[ 2(x + 1) = 2(y - 1) = z + 1 = m. \]

This gives:

\[ x = m - 1, \quad y = \frac{m}{2} + 1, \quad z = m - 1. \]

Now, the line \( L \) passes through both points \( P \) and \( Q \), and we know it is parallel to the direction ratios \( 3 : 1 : 2 \). Substituting the parametric forms into the equations of the line \( L \) and solving for \( k \) and \( m \), we determine the valid points that lie on \( L \).

After solving, it is found that the point:

\[ \left( -\frac{1}{3}, 1, -1 \right) \]

satisfies the equation of \( L \).

Thus, the correct answer is option (2).