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The height of a tower is 'h' and $α$ is the angle of elevation of the top of the tower. On moving a distance $\frac{h}{2}$ towards the tower, the angle of elevation becomes $β$ . What is the value of $cotα - cotβ$ ?

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\frac{1}{2}

\frac{2}{3}

1

2

answer is A.

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Detailed Solution

It is given that the height of a tower is 'h' and

α

is the angle of elevation of the top of the tower. On moving a distance

\frac{h}{2}

towards the tower, the angle of elevation becomes

β

In,

△ ACB,

cotβ = \frac{BC}{AB} = \frac{x}{h}

⇒

cotβ = \frac{x}{h}

⇒

x = hcotβ \dots . (1)

△ ADB,

cotα = \frac{BD}{AB} = \frac{x + \frac{h}{2}}{h}

⇒

hcotα = x + \frac{h}{2}

⇒

x = hcotα - \frac{h}{2} ⟶ (2)

Subtracting

(1)

from

(2),

we get,
⇒

hcotα - \frac{h}{2} - hcotβ = 0

⇒

(cotα - cotβ) = \frac{h}{2}

⇒

cotα - cotβ = \frac{1}{2}

Hence, the correct option is 1.

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