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 $\theta$ 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 $\tan ^{2} \theta$ is equal to

JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Heights and Distances

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{π}{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^{2} θ

is equal to

Answer: (C)

Let

B C = C Q = x

A B = h

, and

P Q = 2 h

.

From C:

tan θ = \frac{h}{x}

and

tan \frac{π}{8} = \frac{2 h}{x}

.

Thus

\frac{t a n θ}{t a n \frac{π}{8}} = \frac{1}{2}

, giving

tan θ = \frac{1}{2} tan \frac{π}{8} = \frac{1}{2} (2 - 1)

.

Therefore

tan^{2} θ = \frac{1}{4} (2 - 1)^{2} = \frac{1}{4} (2 - 22 + 1) = \frac{3 - 2 2}{4}

JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Single correct option