Question

The angle of elevation of the top of the tower from two points P and Q at distance a and b respectively form the base and in the same straight line with it are complementary. Prove that the height of the tower is √ab​ .

Answer 1

The height of the tower is not equal to √ab because a * b is undefined.

Step-by-step explanation:

Let angle of elevation from point P be x, and from point Q be y. Since the angles are complementary, we have x + y = 90°.

Let the height of the tower be h.

In triangle P, using the tangent function, we have tan(x) = h/a. Rearranging this equation, we get h = a * tan(x).

In triangle Q, using the tangent function again, we have tan(y) = h/b. Rearranging this equation, we get h = b * tan(y).

Equating the two expressions for h, we have a * tan(x) = b * tan(y).

Dividing both sides by tan(x) * tan(y), we get a/b = tan(y)/tan(x).

Using the identity tan(x) = 1/tan(90° - x), we have a/b = tan(y)/tan(90° - x).

Simplifying further, we have a/b = cot(x)/cot(y).

Finally, using the identity cot(x) = 1/tan(x), we have a/b = 1/tan(x) * tan(y).

Therefore, a * b = tan(x) * tan(y).

Applying the product-to-sum identity for tangents, we have a * b = (tan(x) + tan(y)) / (1 - tan(x) * tan(y)).

Since x + y = 90°, we have tan(x) + tan(y) = tan(90°) = undefined.

Therefore, 1 - tan(x) * tan(y) = 0.

Dividing both sides of the equation a * b = (tan(x) + tan(y)) / (1 - tan(x) * tan(y)) by 0, we get a * b = undefined.

Since a * b = undefined, we can't conclude that the height of the tower is equal to √ab.