A vertical tower of height h is erected in the middle of a level paddy field. From two points A and B on the same straight horizontal line through the foot of the tower, the angles of elevation of the top of the tower are α and β respectively, where α > β. If the height of the tower is h units, what is a possible distance between the two points A and B (in the same units)?

Introduction / Context:
This question uses basic trigonometry in a height and distance setting with two observation points. The angles of elevation from these points to the top of a vertical tower are given as α and β. You are asked to find the distance between the two points in terms of the tower height h and the trigonometric functions of α and β. Such formulas are commonly used in geometry and aptitude problems involving multiple observations of the same object.

Given Data / Assumptions:

• Vertical tower of height h.
• Angles of elevation from two points A and B are α and β, with α > β.
• A, B and the foot of the tower are on the same straight horizontal line.
• The tower stands perpendicular to the ground.

Concept / Approach:
Let the distances from the foot of the tower to A and B be d₁ and d₂ respectively. Each pair (h, d₁) and (h, d₂) forms a right triangle with angles of elevation α and β. Using tan θ = opposite / adjacent, we express d₁ and d₂ in terms of h, α and β. The distance between A and B is then |d₂ − d₁|, and with α > β we can choose an order that keeps the expression positive.

Step-by-Step Solution:
Let O be the foot of the tower, T the top, and A, B the two ground points. Let OA = d₁ and OB = d₂, with α > β, so A is closer to the tower than B. At A, tan α = h / d₁ ⇒ d₁ = h / tan α = h·cot α. At B, tan β = h / d₂ ⇒ d₂ = h / tan β = h·cot β. Distance between A and B along the line is |d₂ − d₁|. Since β < α, we have d₂ > d₁, so AB = d₂ − d₁. Therefore AB = h·cot β − h·cot α = h(cot β − cot α).
Verification / Alternative check:
You can test with a specific example, say h = 10 units, α = 60° and β = 30°. Compute d₁ = 10·cot 60° and d₂ = 10·cot 30°, then find d₂ − d₁ and confirm that it equals h(cot β − cot α) for these values.

Why Other Options Are Wrong:
The expression h(cot β − cot α) cos(α + β) introduces an extra cosine factor that does not arise from simple right triangle geometry. Expressions involving tan β − tan α or cot α + cot β correspond to different geometric constructions and do not represent the distance between two points on the same line through the base of the tower.

Common Pitfalls:
A frequent error is to mix up which angle corresponds to which point or to mistakenly use tan instead of cot after rearranging. Remember that the point with larger angle of elevation is closer to the tower. Carefully sketching the figure helps keep the relationships clear.

Final Answer:
A possible distance between the two points is h(cot β − cot α).