<p><strong>Ex. 44</strong> Given the line <p>L: \(\frac{x-1}{3} = \frac{y+1}{2} = \frac{z-3}{-1}\)</p> and the plane \(\pi: x - 2y - z = 0\)</p><p><strong>Statement I:</strong> L lies in \(\pi\).</p><p><strong>Statement II:</strong> L is parallel to \(\pi\).</p>
Step-by-Step Solution
Key Concept: A line lies in a plane if all points on the line satisfy the plane equation. A line is parallel to a plane if its direction vector is perpendicular to the plane's normal.
Solution: Parametric form: \(x = 1 + 3r\), \(y = -1 + 2r\), \(z = 3 - r\) Substituting in plane equation: \(1 + 3r - 2(-1 + 2r) - (3 - r) = 0\) \(1 + 3r + 2 - 4r - 3 + r = 0\) \(0 = 0\) (True for all \(r\)) Also, direction ratios of line: \((3, 2, -1)\) Normal to plane: \((1, -2, -1)\) \(3(1) + 2(-2) + (-1)(-1) = 3 - 4 + 1 = 0\) Since the dot product is zero, the line is parallel to the plane. However, checking if line lies in plane: Since the parametric equations satisfy the plane equation for all values of \(r\), the line lies in the plane. ∴ Statement I is true and Statement II is false.
Correct Answer: C