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Grade 11Algebra

A bird flies in a circle on a horizontal plane. An observer stands at a point on the ground. Suppose 60° and 30° are the maximum and the minimum angles of elevation of the bird and that they occur when the bird is at the points P and Q respectively on its path. Let θ be the angle of elevation of the bird when it is a point on the arc of the circle exactly midway between P and Q. Find the numerical value of tan2 θ. (Assume that the observer is not inside the vertical projection of the path of the bird.)

Profile image of Radhika Batra
12 Years agoGrade 11
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1 Answer

Profile image of Jitender Pal
12 Years ago
Hello Student,
Please find the answer to your question
Let A, B, C be the projections of the pts.
P, Q and M on the ground.
ATQ, ∠POA = 60°
∠QOB = 30°
∠MOC = θ
Let h be the ht of circle from ground, then
AP = CM = BQ = h
Let OA = x and AB = d(diameter of the projection of the circle on ground with C1 as centre).
235-1550_12345.png
Now in ∆POA, tan 60° = h/x ⇒ x = h/√3 . . . . . . . . . . . . . (1)
In ∆QBO, tan 30° = h/x + d ⇒ x + d = h√3
⇒ d = h√3 – h/√3 = 2h/√3 . . . . . . . . . . . . . . . . (2)
In ∆OMC, tan θ = h/OC
⇒ tan2 θ = h2/OC2 = h2/OC21 + C1 C2 = h2/(x + d/2)2 + (d/2)2
= \frac{h^2}{(\frac{h}{\sqrt{3}}+\frac{h}{\sqrt{3}})^2+(\frac{h}{\sqrt{3}})^2}[ Using (1) and (2)
= \frac{h^2}{\frac{4h^2}{3}+\frac{h^2}{3}}= 3/5

Thanks
Jitender Pal
askIITians Faculty

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