Question

The point of intersection of the lines \( \frac{x - 1}{3} = \frac{y - 2}{-3} = \frac{z - 3}{4} \) and \( \frac{x - 4}{5} = \frac{y - 1}{2} = \frac{z - 1}{-2} \) is:

• A. \( (0, 0, 0) \)
• B. \( (1, 1, 1) \)
• C. \( (-1, -1, -1) \)
• D. \( (1, 2, 3) \)

Hint:

To find the point of intersection of two lines, equate their parametric equations and solve for the parameters.

Answer 1

The Correct Option is A

To find the point of intersection, equate the parametric equations of both lines. For the first line: \[ \frac{x - 1}{3} = \frac{y - 2}{-3} = \frac{z - 3}{4} = t \] This gives the parametric equations: \[ x = 3t + 1, \quad y = -3t + 2, \quad z = 4t + 3 \] For the second line: \[ \frac{x - 4}{5} = \frac{y - 1}{2} = \frac{z - 1}{-2} = s \] This gives the parametric equations: \[ x = 5s + 4, \quad y = 2s + 1, \quad z = -2s + 1 \] Equate the expressions for \( x \), \( y \), and \( z \) from both lines to solve for \( t \) and \( s \). Solving these gives the point of intersection \( (0, 0, 0) \).