<p>If the line \(\dfrac{x-3}{2} = \dfrac{y+2}{-1} = \dfrac{z+4}{3}\) lies in the plane \(lx + my - z = 9\), then \(l^2 + m^2\) is equal to</p>
Step-by-Step Solution
Key Concept: For a line to lie entirely in a plane, two conditions must be satisfied: (1) the direction vector must be perpendicular to the normal vector of the plane, and (2) any point on the line must satisfy the plane equation.
Step 1: Identify the direction vector and a point on the line. Direction vector: d = (2, -1, 3) Point on line: P(3, -2, -4) Step 2: Apply perpendicularity condition. Normal vector of plane: n = (l, m, -1) For line to lie in plane: d · n = 0 2l + (-1)m + 3(-1) = 0 2l - m - 3 = 0 ... (equation 1) Step 3: Apply point condition. Point P(3, -2, -4) must satisfy the plane equation lx + my - z = 9 l(3) + m(-2) - (-4) = 9 3l - 2m + 4 = 9 3l - 2m = 5 ... (equation 2) Step 4: Solve the system of equations. From equation 1: m = 2l - 3 Substitute into equation 2: 3l - 2(2l - 3) = 5 3l - 4l + 6 = 5 -l = -1 l = 1 Therefore: m = 2(1) - 3 = -1 Step 5: Calculate l^2 + m^2. l^2 + m^2 = 1^2 + (-1)^2 = 1 + 1 = 2 ∴ Answer: 2
Correct Answer: A