Let a vertical tower AB of height 2h stands on a horizontal ground. Let from a point P on the ground a man can see upto height h of the tower with an angle of elevation $2\mathrm{\alpha }$. When from P, he moves a distance d in the direction of $\overrightarrow{\mathrm{AP}}$, he can see the top B of the tower with an angle of elevation $\mathrm{\alpha }$. If $\mathrm{d}=\sqrt{7}\mathrm{h}$, then $\mathrm{tan\alpha }$ is equal to

Let a vertical tower AB of height 2h stand on a horizontal ground. From a point P on the ground, a man can see up to height h of the tower with an angle of elevation 2α. When he moves a distance d in the direction of AP, he can see the top B of the tower with an angle of elevation α. It is also given that d = 7h. We need to find tanα.

Step 1: Analyzing Triangle APM

In the first scenario, the man can see up to height h of the tower. The angle of elevation is 2α, and the distance from point P to the point directly below height h is x.

Using trigonometric relations in triangle APM, we have:

tan(2α) = h / x    

Thus, equation (i) becomes:

tan(2α) = h / x  ... (i)    

Step 2: Analyzing Triangle AQB

Now, the man moves a distance d = 7h towards the point A and can see the top B of the tower. The distance between point Q (where the man now stands) and the base of the tower is x + d = x + 7h. The angle of elevation is now α.

Using trigonometric relations in triangle AQB, we get:

tan(α) = 2h / (x + d)    

Substituting d = 7h, we get:

tan(α) = 2h / (x + 7h)    

Thus, equation (ii) becomes:

tan(α) = 2h / (x + 7h)  ... (ii)    

Step 3: Solving the System of Equations

Now, we have two equations:

• tan(2α) = h / x ... (i)
• tan(α) = 2h / (x + 7h) ... (ii)

From equation (i), we can express x in terms of h and tan(2α):

x = h / tan(2α)    

Substituting this expression for x in equation (ii), we get:

tan(α) = 2h / (h / tan(2α) + 7h)    

Simplifying this, we get:

tan(α) = 2h / (h(1 / tan(2α) + 7))    

Now, multiply both sides of the equation by tan(2α) and simplify further:

tan(α) * tan(2α) = 2 / (1 + 7 * tan(2α))    

Let t = tan(α) and use the double angle identity for tangent:

tan(2α) = (2 * tan(α)) / (1 - tan^2(α))    

Substituting this into the equation, we get a quadratic equation in terms of t:

t^2 - (2√7) * t + 3 = 0    

Step 4: Solving the Quadratic Equation

Now, solve the quadratic equation for t (which is tan(α)):

t = [2√7 ± √(4(7) - 4 * 3)] / 2    

Simplifying the discriminant:

t = [2√7 ± √(28 - 12)] / 2    

t = [2√7 ± √16] / 2            

t = [2√7 ± 4] / 2    

Thus, we get two possible values for t (which corresponds to tan(α)):

t = (2√7 + 4) / 2 = √7 + 2    

or

t = (2√7 - 4) / 2 = √7 - 2    

Final Answer

The value of tan(α) is either √7 + 2 or √7 - 2.

Thus, the correct answer is tan(α) = √7 - 2.