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The angle of elevation of a tower from a point $A$ due south of it, is $x$ , from a point $B$ due east of $A$ , is $y$ . If $AB = l,$ then the height $h$ of the tower is given by

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\frac{l}{\sqrt{\cot^{2} y - \cot^{2} x}}

\frac{l}{\sqrt{\tan^{2} y - \tan^{2} x}}

\frac{2 l}{\sqrt{\cot^{2} y - \cot^{2} x}}

None of these

answer is A.

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Detailed Solution

It is given that the angle of elevation of a tower from a point

A

due south of it, is

x

, from a point

B

due east of

A

, is

y

.
Let

OP

be the tower of height

h

In right angled

△ OAP,

∠ OAP = x

⇒

\frac{OA}{h} = cotx

⇒

OA = hcotx

In right angled

△ OBP

∠ OBP = y

⇒

\frac{OB}{h} = coty

⇒

OB = hcoty

In right angled

△ OAB

,
By pythagoras theorem,

{AB}^{2} + {OA}^{2} = {OB}^{2}

⇒

l^{2} + h^{2} \cot^{2} x = h^{2} \cot^{2} y

⇒

h^{2} (\cot^{2} y - \cot^{2} x) = l^{2}

⇒

h = \frac{l}{\sqrt{\cot^{2} y - \cot^{2} x}}

Hence, the correct option is 1.

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The angle of elevation of a tower from a point A due south of it, is x, from a point B due east of A, is y. If AB=l, then the height h of the tower is given by

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The angle of elevation of a tower from a point $A$ due south of it, is $x$ , from a point $B$ due east of $A$ , is $y$ . If $AB = l,$ then the height $h$ of the tower is given by