<p>The line \(\dfrac{x-3}{2} = \dfrac{y+2}{-1} = \dfrac{z+4}{3}\) lies in the plane \(lx + my - z = 9\). Find \(l^2 + m^2\).</p>
Step-by-Step Solution
Key Concept: A line lies in a plane if and only if: (1) the direction vector of the line is perpendicular to the normal vector of the plane, and (2) any point on the line satisfies the plane equation. Use both conditions simultaneously to find l and m.
Step 1: Identify the line's direction vector and a point on it. Direction vector: d = (2, -1, 3) Point on line: P = (3, -2, -4) Step 2: For the line to lie in the plane lx + my - z = 9, the direction vector must be perpendicular to the plane's normal vector n = (l, m, -1). d · n = 0 2l - m(1) + 3(-1) = 0 2l - m - 3 = 0 Equation (1): 2l - m = 3 Step 3: The point P(3, -2, -4) must satisfy the plane equation lx + my - z = 9. l(3) + m(-2) - (-4) = 9 3l - 2m + 4 = 9 Equation (2): 3l - 2m = 5 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: B
Correct Answer: B