A flagstaff is placed on top of a building. The flagstaff and building subtend equal angles at a point on level ground which is 200 m away from the foot of the building. If the height of the flagstaff is 50 m and the height of the building is h, which of the following is true?

Answer: Option C

Let AD be the flagstaff and CD be the building.
Assume that the flagstaff and building subtend equal angles at point B.
Given that AD = 50 m, CD = h and BC = 200 m
Let ∠ABD = $$\theta$$, ∠DBC = $$\theta$$   (∵ flagstaff and building subtend equal angles at a point on level ground).
Then, ∠ABC = 2$$\theta$$
$$\begin{array}{rl}& \text{From}\text{the}\text{right}\Delta BCD,\\ & \tan \theta =\frac{DC}{BC}\\ & =\frac{h}{200}......\left(1\right)\\ & \text{From}\text{the}\text{right}\Delta BCA,\\ & \tan 2\theta =\frac{AC}{BC}\\ & =\frac{AD+DC}{200}\\ & =\frac{50+h}{200}\end{array}$$
$$\Rightarrow \frac{2\tan \theta }{1-{\tan }^2\theta }=\frac{50+h}{200}$$      $$\left(\because \tan \left(2\theta \right)=\frac{2\tan \theta }{1-{\tan }^2\theta }\right)$$
$$\frac{2\left(\frac{h}{200}\right)}{1-\frac{h^2}{{200}^2}}=\frac{50+h}{200}$$      (∵ substituted value of tan $$\theta$$ from eq:1)
$$\begin{array}{rl}& \Rightarrow 2h=\left(1-\frac{h^2}{{200}^2}\right)\left(50+h\right)\\ & \Rightarrow 2h=50+h-\frac{50h^2}{{200}^2}-\frac{h^3}{{200}^2}\end{array}$$
$$\Rightarrow 2\left({200}^2\right)h=50{\left(200\right)}^2+$$     $$h{\left(200\right)}^2-$$   $$50h^2-h^3$$
(∵ multiplied LHS and RHS by $${200}^2$$ )

h³ + 50h² + (200)²h - (200)²50 = 0