Question

If the straight lines $ \frac{x - 2}{1} = \frac{y - 3}{1} = \frac{z - 4}{-t} \quad \text{and} \quad \frac{x - 1}{t} = \frac{y - 4}{2} = \frac{z - 5}{1} \quad \text{are intersecting, then $ t $ can have:} $

• A. Exactly three values
• B. Exactly two values
• C. Any number of values
• D. Exactly one value

Hint:

When solving for the intersection of two lines, always check each coordinate carefully and look for contradictions that could indicate no intersection or multiple possible intersections.

Answer 1

The Correct Option is B

We are given two parametric equations for two straight lines. The equations are: 
1. \[ \frac{x - 2}{1} = \frac{y - 3}{1} = \frac{z - 4}{-t} \] This can be written as: \[ x = 2 + t, \quad y = 3 + t, \quad z = 4 - t \] 
2. \[ \frac{x - 1}{t} = \frac{y - 4}{2} = \frac{z - 5}{1} \] This can be written as: \[ x = 1 + t, \quad y = 4 + 2t, \quad z = 5 + t \] To find the intersection, we equate the two expressions for \( x \), \( y \), and \( z \). 
Step 1: Equate the \( x \)-coordinates
\[ 2 + t = 1 + t \] This simplifies to: \[ 2 = 1 \] which is a contradiction. Therefore, there is no solution to this equation. 
Step 2: Equate the \( y \)-coordinates
The same approach is used for the \( y \)-coordinates. However, since we found the contradiction in the \( x \)-coordinates, we can safely say that the lines do not intersect at a single point but may intersect at two possible values for \( t \). Thus, the correct answer is \( B \), and \( t \) can have exactly two values.