<p><strong>Ex. 45</strong> <strong>Statement I:</strong> Line \(\frac{x-1}{3} = \frac{y-2}{11} = \frac{z+1}{11}\) lies in the plane \(11x - 3z - 14 = 0\).</p><p><strong>Statement II:</strong> A straight line lies in a plane, if the line is parallel to the plane and a point of the line is in the plane.</p>
Step-by-Step Solution
Key Concept: A line lies in a plane if and only if (1) a point on the line satisfies the plane equation, and (2) the direction vector of the line is perpendicular to the normal of the plane.
Solution: For Statement I: Check point \((1, 2, -1)\) on the line: \(11(1) - 3(-1) - 14 = 11 + 3 - 14 = 0\) Point lies on plane. ✓ Check if direction vector is parallel to plane: Direction vector: \((3, 11, 11)\) Normal to plane: \((11, 0, -3)\) \(3(11) + 11(0) + 11(-3) = 33 + 0 - 33 = 0\) ✓ ∴ Statement I is true. For Statement II: The condition stated is exactly the criterion for a line to lie in a plane. ∴ Statement II is true and is the correct explanation of Statement I.
Correct Answer: A