Question

Let AB and PQ be two vertical poles, 160 m apart from each other. Let C be the middle point of B and Q, which are feet of these two poles. Let $\frac \pi 8$ and $θ$ be the angles of elevation from C to P and A, respectively. If the height of pole PQ is twice the height of pole AB, then $tan^2θ$ is equal to

• A. $\frac {3-2\sqrt 2}{2}$
• B. $\frac {3+\sqrt 2}{4}$
• C. $\frac {3-2\sqrt 2}{2}$
• D. $\frac {3-\sqrt 2}{4}$

Answer 1

Answer: (C)

$\frac {3-2\sqrt 2}{2}$

Answer 2

$\frac {l}{80} = tan\ θ$ ..... (i)

$\frac {2l}{80} = tan \frac \pi8$ ...... (ii)

From (i) and (ii)

$\frac 12 = \frac {tan\ θ}{tan \frac {\pi}{8}}$

$⇒ tan^2θ = \frac 14 tan^2\frac \pi 8$

$⇒ tan^2θ = \frac {\sqrt 2 - 1}{4(\sqrt 2 + 1)}$

$⇒ tan^2θ =\frac {3-2\sqrt 2}{2}$

So, the correct option is (C): $\frac {3-2\sqrt 2}{2}$