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Offline Centres

State true or false.

A round balloon of radius a subtends an angle α at the eye of the observer while the

Angle of elevation of its centre is $β$ then height of the center of the balloon is

$r sin β \csc \frac{α}{2}$ .

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True

False

answer is A.

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Detailed Solution

Assume that
Centre of the balloon = O
The eye of the observer = P
The tangents to the round balloon = PA and PB
The horizontal line = PX
And

CQ ⊥ PQ

Given,

∠ APB = α

Thus,

∠ CPA = ∠ CPB = \frac{α}{2}

And

∠ CPQ = β

Radius of balloon,

CA = CB = a

C:UserssoftberryDownloadsP0041 JPEGQ.86.jpg

Now,
The height of the centre C = h
In right triangle

∆

CBP, from the property of trigonometry

= \sin \frac{α}{2} = \frac{BC}{CP}

\Rightarrow \sin \frac{α}{2} = \frac{a}{CP}

\Rightarrow CP = \frac{a}{\sin \frac{α}{2}}

CP = a \csc (\frac{α}{2}) \dots

(1)
Again in right triangle

∆

CPQ,

= \sin β = \frac{CQ}{CP}

\Rightarrow CQ = CPsin β

\Rightarrow CQ = acsc (\frac{α}{2}) \sin β

Therefore, the given statement is true.
Correct option is 1.

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