Question

From the top of a tower \(h\) meter high, the angle of depression of two objects, which lie on either side of it are \( \alpha \) and \( \beta \). The distance between the two objects is :

• A. \( h(\cot\alpha + \cot\beta) \)
• B. \( h(\cot\alpha - \cot\beta) \)
• C. \( h(\tan\alpha + \tan\beta) \)
• D. \( h(\tan\alpha - \tan\beta) \)

Hint:

% Option (A) Draw the tower (height \(h\)) and the two objects O1 and O2 on opposite sides at ground level. % Option (B) The angles of depression (\(\alpha, \beta\)) from the top of the tower to O1 and O2 are equal to the angles of elevation from O1 and O2 to the top of the tower, respectively. % Option (C) Let \(d_1\) be the distance from the foot of the tower to O1, and \(d_2\) be the distance to O2. % Option (D) In the right triangle for O1: \(\tan\alpha = h/d_1 \implies d_1 = h/\tan\alpha = h\cot\alpha\). % Option (E) In the right triangle for O2: \(\tan\beta = h/d_2 \implies d_2 = h/\tan\beta = h\cot\beta\). % Option (F) Total distance between objects = \(d_1 + d_2 = h\cot\alpha + h\cot\beta = h(\cot\alpha + \cot\beta)\).

Answer 1

The Correct Option is A

Concept: This problem involves angles of depression and basic trigonometry in right-angled triangles. The angle of depression from an observer to an object is the angle between the horizontal line from the observer and the line of sight to the object, when the object is below the horizontal line. Step 1: Draw a diagram Let T be the top of the tower and F be its foot. The height of the tower TF = \(h\). Let the two objects be O1 and O2, on either side of the foot of the tower, in line with the foot. Let the horizontal line from T be TX. Angle of depression of O1 is \(\angle XTO_1 = \alpha\). Angle of depression of O2 is \(\angle XTO_2 = \beta\). Since TX is parallel to the ground \(FO_1O_2\): \(\angle TO_1F = \angle XTO_1 = \alpha\) (alternate interior angles). \(\angle TO_2F = \angle XTO_2 = \beta\) (alternate interior angles). We have two right-angled triangles: \(\triangle TFO_1\) (right-angled at F) and \(\triangle TFO_2\) (right-angled at F). The distance between the two objects is \(O_1O_2 = FO_1 + FO_2\). Step 2: Calculate \(FO_1\) using \(\triangle TFO_1\) In right-angled \(\triangle TFO_1\): Angle at \(O_1\) is \(\alpha\). Side opposite to \(\alpha\) (height) is TF = \(h\). Side adjacent to \(\alpha\) (base) is \(FO_1\). We can use \(\tan\alpha = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{TF}{FO_1} = \frac{h}{FO_1}\). So, \(FO_1 = \frac{h}{\tan\alpha} = h\cot\alpha\). Step 3: Calculate \(FO_2\) using \(\triangle TFO_2\) In right-angled \(\triangle TFO_2\): Angle at \(O_2\) is \(\beta\). Side opposite to \(\beta\) (height) is TF = \(h\). Side adjacent to \(\beta\) (base) is \(FO_2\). We can use \(\tan\beta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{TF}{FO_2} = \frac{h}{FO_2}\). So, \(FO_2 = \frac{h}{\tan\beta} = h\cot\beta\). Step 4: Calculate the distance between the objects \(O_1O_2\) Distance \(O_1O_2 = FO_1 + FO_2\). Substitute the expressions for \(FO_1\) and \(FO_2\): \(O_1O_2 = h\cot\alpha + h\cot\beta\). Factor out \(h\): \(O_1O_2 = h(\cot\alpha + \cot\beta)\). This matches option (1).