Question

The line \(\ell\) passes through the point \((2,1,1)\) and is parallel to the plane \(x+y+2z = 18\). If line \(\ell\) intersects the line \(\frac{x+2}{3} = \frac{y+1}{-1} = \frac{z-2}{1}\), then equation of the line \(\ell\) is...

• A. \(\frac{x-2}{3} = \frac{y-1}{1} = \frac{z-1}{-2}\)
• B. \(\frac{x-3}{1} = \frac{y-2}{3} = \frac{z-2}{-2}\)
• C. \(\frac{x-2}{3} = \frac{y-1}{-1} = \frac{z-1}{-1}\)
• D. \(\frac{x-3}{1} = \frac{y-4}{3} = \frac{z+1}{-2}\)

Hint:

Find the point on the given line using the parallel condition, then join it to the point.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A line parallel to the plane \(x + y + 2z = 18\) has a direction vector perpendicular to the plane normal \((1, 1, 2)\). The line must also meet the given line.

Step 2: Key Formula or Approach:
A general point on the given line: \(P = (-2 + 3s,\ -1 - s,\ 2 + s)\). The direction from \((2, 1, 1)\) to \(P\) is \((-4 + 3s,\ -2 - s,\ 1 + s)\).

Step 3: Detailed Explanation:
Parallel to the plane: \((-4 + 3s) + (-2 - s) + 2(1 + s) = 0\), so \(-4 + 4s = 0\) and \(s = 1\).
Then \(P = (1, -2, 3)\) and the direction is \((-1, -3, 2)\), which is proportional to \((1, 3, -2)\).
The line is \(\frac{x - 2}{1} = \frac{y - 1}{3} = \frac{z - 1}{-2}\).
Option D, \(\frac{x - 3}{1} = \frac{y - 4}{3} = \frac{z + 1}{-2}\), has the same direction and passes through \((3, 4, -1) = (2, 1, 1) + (1, 3, -2)\), so it is the same line.

Final Answer:
The line is option (D). \[ \boxed{\frac{x-3}{1}=\frac{y-4}{3}=\frac{z+1}{-2}} \]