Question

The point of intersection of the lines \( \frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \) and \( \frac{x - 5}{2} = \frac{y - 2}{1} = \frac{z - 5}{2} \)

• A. \( (1, -2, 0) \)
• B. \( (3, 1, 0) \)
• C. \( (-2, -5, -7) \)
• D. None of these

Hint:

If two lines are not parallel and do not intersect, then they are skew lines. Check if the system of equations has no solution.

Answer 1

The Correct Option is D

We are given two parametric equations for two lines. We will first convert the equations into parametric form, and then solve for the point of intersection. 1. Equation of first line: \[ x = 1 + t, \quad y = 1 + 2t, \quad z = 2 + 3t \] where \( t \) is the parameter for the first line. 2. Equation of second line: \[ x = 5 + 2s, \quad y = 2 + s, \quad z = 5 + 2s \] where \( s \) is the parameter for the second line. Now, equate the expressions for \( x \), \( y \), and \( z \) from both lines: \[ 1 + t = 5 + 2s \quad \Rightarrow \quad t - 2s = 4 \quad \cdots (1) \] \[ 1 + 2t = 2 + s \quad \Rightarrow \quad 2t - s = 1 \quad \cdots (2) \] \[ 2 + 3t = 5 + 2s \quad \Rightarrow \quad 3t - 2s = 3 \quad \cdots (3) \] Solve this system of equations to find \( t \) and \( s \). After solving, we find that there is no common solution that satisfies all three equations, meaning the two lines do not intersect at any point. Hence, the correct answer is (D) None of these.