MathematicsA person observes the top P of a vertical tower OP of height h from a station S1 and finds that β1 is the angle of elevation. He moves in a horizontal plane to second station S2 and finds that ∠PS2S1 is γ1 and the angle subtended by S2S1 at P is δ1 and the angle of elevation is β2. He moves again to a third station S3 such that S3S2=S2S1, ∠PS3S2=γ2 and the angle subtended by S3S2 at P is δ2. then sin⁡γ1sin⁡β1sin⁡δ1=sin⁡γ2sin⁡β2sin⁡δ2=hS1S2

A person observes the top P of a vertical tower OP of height h from a station $S_{1}$ and finds that $β_{1}$ is the angle of elevation. He moves in a horizontal plane to second station S2 and finds that $∠ P S_{2} S_{1}$ is $γ_{1}$ and the angle subtended by $S_{2} S_{1}$ at P is $δ_{1}$ and the angle of elevation is β2. He moves again to a third station $S_{3}$ such that $S_{3} S_{2} = S_{2} S_{1}, ∠ P S_{3} S_{2} = γ_{2}$ and the angle subtended by $S_{3} S_{2}$ at P is $δ_{2}$ . then

$\frac{\sin γ_{1} \sin β_{1}}{\sin δ_{1}} = \frac{\sin γ_{2} \sin β_{2}}{\sin δ_{2}} = \frac{h}{S_{1} S_{2}}$

A
True
B
False

Solution:

Sine rule in given as,

\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}

Here,
a, b, c

\to

sides of a triangle
A, B, C

\to

Angles (lying opposite to sides a, b, c respectively.)
Use sine rule in the triangles

P S_{1} S_{2}

and

P S_{2} S_{3}

with the involvement of sides

P S_{1} S_{1}

and

S_{2} S_{1}

for triangle

P S_{1} S_{2}

and sides

P S_{2} S_{2}

and

S_{3} S_{2}

for triangle

P S_{2} S_{3}

.
It is known that,

sinθ = \frac{Perpendicular}{Hypotaneous}

h = P S_{1} ​ \sin β_{1} ​

\Rightarrow P S_{1} ​ = h cosec β_{1} ​ .

From

Δ {PS}_{1} ​ S_{2}

by sine formula,

\frac{P S_{1} ​​}{\sin γ_{1} ​​​} = \frac{S_{1} ​ S_{2} ​}{\sin δ_{1} ​​​}

\Rightarrow \frac{h ​​}{\sin β_{1} \sin γ_{1} ​​​} = \frac{S_{1} ​ S_{2} ​}{\sin δ_{1} ​​​} ​

……. (1)

\Rightarrow \frac{h ​​}{S_{1} ​ S_{2} ​​} = \frac{\sin β_{1} \sin γ_{1} ​}{\sin δ_{1} ​​​}

In the same way,

\frac{h ​​}{S_{2} ​ S_{3} ​​} = \frac{\sin β_{2} \sin γ_{2} ​}{\sin δ_{2} ​​​}

But

S_{2} ​ S_{3} = S_{1} ​ S_{2}

(given)

\frac{\sin β_{1} \sin γ_{1} ​}{\sin δ_{1} ​​​} ​​

= \frac{\sin β_{2} \sin γ_{2} ​}{\sin δ_{2} ​​​}

= \frac{h ​​}{S_{1} ​ S_{2} ​​}

Option 1 is Correct.

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