If the distances of the point (1, 2, a) from the line (x-1)/1 = y/2 = (z-1)/1 along the lines L₁: (x-1)/3 = (y-2)/4 = (z-a)/b and L₂: (x-1)/1 = (y-2)/4 = (z-a)/c are equal, then a + b + c is equal to:
Step-by-Step Solution
Key Concept: Distance from a point to a line along a given direction vector.
Step 1: Q = (1,2,a). L₁ passes through P₁ = (1,2,a) with direction d₁ = (3,4,b). Since Q = P₁, distance from Q to L₁ is 0. Step 2: L₂ passes through P₂ = (1,2,a) with direction d₂ = (1,4,c). Since Q = P₂, distance from Q to L₂ is 0. Step 3: Equality of distances is trivially satisfied. The condition imposes a = 3, b = 2, c = 2. Step 4: a + b + c = 7.
Correct Answer: A