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The top of hill observed from the top and bottom of a building of height $h$ is at the angles of elevation $p$ and $q$ respectively. The height of the hill is

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\frac{hcotp}{cotp - cotq}

\frac{hcotq}{cotp - cotq}

\frac{hcotp}{cotq - cotp}

\frac{hcotq}{cotP + cotq}

answer is A.

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Detailed Solution

It is given the top of hill observed from the top and bottom of a building which has height

h

is at the angles of elevation

p

and

q

respectively.
According to the given data, the figure has been drawn below.
Question Image

Here

h_{1}

is the height of the hill, and CD=AB=d.
In

△

CDE ,
∠ECD = p
Now,

tan θ = P e r p e n d i c u l a r b a s e

tanp = \frac{ED}{CD}

= \frac{h_{1} - h}{d}

….(1)
In

Δ A B E

tan θ = P e r p e n d i c u l a r b a s e

tanq = \frac{EB}{AB}

= \frac{h_{1}}{d}

…..(2)
Dividing the quation (1) by equation (2), we get,

\frac{tanp}{tanq} = \frac{h_{1} - h}{h_{1}}

\Rightarrow \frac{tanp}{tanq} = 1 - \frac{h}{h_{1}}

\Rightarrow \frac{h}{h_{1}} = 1 - \frac{tanp}{tanq}

\Rightarrow \frac{h_{1}}{h} = \frac{cotp}{cotp - cotq}

∵ 1 tan θ = cot θ

\Rightarrow h_{1} = \frac{hcotp}{cotp - cotq}

So the height of the hill is

\frac{hcotp}{cotp - cotq}

.
So the correct option is 1.

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