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A tower stands on the edge of the circular lake ABCD. the foot

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A tower stands on the edge of the circular lake ABCD. the foot of the tower is at D and the angles of elevation of its top at A, B, C are respectively $α, β$ and $γ$ . If the angles BAC, BCA are each $θ$ , show that $2 cos θ cot β = cot α + cot γ$ .

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A tower stands on the edge of the circular lake ABCD. the foot of the tower is at D and the angles of elevation of its top at A, B, C are respectively

α, β

and

γ

. If the angles BAC, BCA are each

θ

, show that

2 cos θ cot β = cot α + cot γ

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Text solutionVerified

let h be the height of the tower. Then

D A = h cot α, D B = h cot β

and

D C = h cot γ

.
Also, it is given that

α B A C = ∠ BC A = θ

so that BA = BC and hence

A C = L A + L C = A B cos θ + BC cos θ

A C = 2 A B cos θ

...(1)
Again ABCD is a cyclic quadrilateral and hence by Ptolemy's theorem, we get

A B . C D + A D . BC = A C . B D

2 cos θ co tβ = co t α + co t γ

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Question Text	A tower stands on the edge of the circular lake ABCD. the foot of the tower is at D and the angles of elevation of its top at A, B, C are respectively $α, β$ and $γ$ . If the angles BAC, BCA are each $θ$ , show that $2 cos θ cot β = cot α + cot γ$ .
Answer Type	Text solution:1
Upvotes	150

A tower stands on the edge of the circular lake ABCD. the foot of the tower is at D and the angles of elevation of its top at A, B, C are respectively α,β and γ. If the angles BAC, BCA are each θ, show that 2cosθcotβ=cotα+cotγ.

Text solutionVerified

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A tower stands on the edge of the circular lake ABCD. the foot of the tower is at D and the angles of elevation of its top at A, B, C are respectively $α, β$ and $γ$ . If the angles BAC, BCA are each $θ$ , show that $2 cos θ cot β = cot α + cot γ$ .