Question

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

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

Hint:

Write both lines in parametric form and equate coordinates.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Two lines intersect when there are parameter values at which their points coincide. Write general points of each line.

Step 2: Key Formula or Approach:
Line 1: \((3 + s,\ 5 + 2s,\ 1 - s)\). Line 2: \((4 + 2t,\ 2 - t,\ 4 + 2t)\).

Step 3: Detailed Explanation:
Equate: \(3 + s = 4 + 2t\) (i), \(5 + 2s = 2 - t\) (ii), \(1 - s = 4 + 2t\) (iii).
From (i): \(s = 1 + 2t\). Put into (iii): \(1 - 1 - 2t = 4 + 2t\), so \(t = -1\) and \(s = -1\).
Check (ii): \(5 - 2 = 3\) and \(2 + 1 = 3\). It holds.
Point: \((3 - 1,\ 5 - 2,\ 1 + 1) = (2, 3, 2)\).

Final Answer:
The point of intersection is \((2, 3, 2)\), option (A). \[ \boxed{(2,3,2)} \]