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Circles

Question
CBSEENMA10009640

At a point A, 20 metres above the level of water in a lake, the angle of elevation of a cloud is 30°. The angle of depression of the reflection of the cloud in the lake, at A is 60°. Find the distance of the cloud from A.

Solution

Let BE be the surface of the lake. Suppose C be the position of the cloud and C' be its reflection in the lake.
Then, AB = 20 m
Let CD = x m. Then,
CE =C'E = ( x + 20)
In ΔACD,
tan space 30 degree space equals space CD over AD
rightwards double arrow space fraction numerator 1 over denominator square root of 3 space end fraction space equals space straight x over AD

rightwards double arrow AD space equals space straight x square root of 3 space space.. space left parenthesis straight i right parenthesis

Also comma

sin space 30 degree space equals space CD over AC

rightwards double arrow space 1 half space equals space straight x over AC
rightwards double arrow space AC space equals 2 straight x space.... space left parenthesis ii right parenthesis

Now comma space in space ΔADC apostrophe comma
tan space 60 degree space equals space fraction numerator straight C apostrophe straight D over denominator AD end fraction
rightwards double arrow space square root of 3 space equals space fraction numerator left parenthesis straight x space plus 20 right parenthesis space plus space 20 over denominator AD end fraction
rightwards double arrow space AD space equals space fraction numerator left parenthesis straight x space plus 40 right parenthesis over denominator square root of 3 end fraction space... space left parenthesis iii right parenthesis
From space left parenthesis straight i right parenthesis space and space left parenthesis iii right parenthesis comma space we space get
straight x square root of 3 space equals space fraction numerator left parenthesis straight x space plus space 40 right parenthesis over denominator square root of 3 end fraction
rightwards double arrow space 3 straight x space equals space straight x space plus 40
rightwards double arrow space 2 straight x space equals space 40
rightwards double arrow space straight x space equals space 20

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