<p>If PQ be a vertical tower subtending angles α, β and γ at the points A, B and C respectively on the line in the horizontal plane through the foot D of tower and on the same side of it, then BC cot α - CA cot β + AB cot γ is equal to</p>
Step-by-Step Solution
Key Concept: Set up the relationship between the angles subtended by the tower at different points on the ground using the cotangent formula: cot θ = (horizontal distance)/(height). Then use properties of collinear points to establish a linear dependence condition.
<p><strong>Step 1:</strong> Set up the problem. Let PQ be a vertical tower of height h, with foot at D. Points A, B, C lie on a horizontal line through D (all on the same side). The angles subtended by tower PQ at A, B, C are α, β, γ respectively.</p><p><strong>Step 2:</strong> Express cotangent relationships. Let DA = a, DB = b, DC = c. Then:<br/>cot α = a/h, cot β = b/h, cot γ = c/h</p><p><strong>Step 3:</strong> Express distances in terms of cotangents.<br/>a = h cot α, b = h cot β, c = h cot γ</p><p><strong>Step 4:</strong> Use the collinearity condition. Since A, B, C are collinear points on the same line through D, the signed distances must satisfy the condition that these three points lie on a straight line. Without loss of generality, assume the order as A, B, C on the line.</p><p><strong>Step 5:</strong> Apply the constraint. If the points are collinear with A, B, C in order, then either:<br/>AC = AB + BC, or other arrangements depending on order.</p><p><strong>Step 6:</strong> For any three collinear points, we have the identity:<br/>BC · cot α - CA · cot β + AB · cot γ<br/>= BC · (a/h) - CA · (b/h) + AB · (c/h)<br/>= (1/h)[BC · a - CA · b + AB · c]</p><p><strong>Step 7:</strong> Use Menelaus' theorem or direct geometric analysis. For three collinear points on a line through the foot of the tower, the expression BC · cot α - CA · cot β + AB · cot γ represents a linear combination. By the principle of collinearity (applying signed lengths), this sum identically equals zero.</p><p><strong>Step 8:</strong> Verification: The expression BC cot α - CA cot β + AB cot γ = 0 is a fundamental identity for any three collinear points with respect to a perpendicular from an external point.</p><p><strong>∴ Answer:</strong> A</p>
Correct Answer: A