# 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.)

Jitender Pal
9 years ago
Hello Student,
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).
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