Multiple choice

The angle of elevation of the top of a tower as seen from two points $A$ & $B$ situated the same line and at distance '$p$' and '$q$' respectively from the foot of the tower are complementary, then height of the tower is

  1. $pq$
  2. $\displaystyle \frac{p}{q}$
  3. $\displaystyle \sqrt{pq}$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let height be h. tan(theta) = h/p and tan(90-theta) = h/q. Thus, cot(theta) = h/q. Since tan(theta) * cot(theta) = 1, (h/p) * (h/q) = 1, so h^2 = pq, h = sqrt(pq).

AI explanation

Let the height of the tower be h and the complementary angles of elevation be theta and (90 degrees minus theta). Using the trigonometric ratio tan(theta) = h/p and tan(90 degrees minus theta) = h/q, which simplifies to cot(theta) = h/q. Multiplying the two equations gives tan(theta) multiplied by cot(theta) = (h/p) multiplied by (h/q), leading to 1 = h^2/pq. Solving for h yields the height of the tower as sqrt(pq).

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