Problem 69 - A Complete Resource Book in Mathematics for JEE Main

Question

If the angles of elevation of the top of a tower from three collinear points \(A, B\) and \(C\), on a line leading to the foot of the tower, are \(30^{\circ}, 45^{\circ}\) and \(60^{\circ}\) respectively, then the ratio, \(A B: B C\), is: (A) \(\sqrt{3}: \sqrt{2}\) (B) \(1: \sqrt{3}\) (C) \(2: 3\) (D) \(\sqrt{3}: 1\)

Step-by-Step Solution

Answer

The ratio is \(\sqrt{3} : 1\), option (D).

Step 1: Understand the Problem

In the problem, we are dealing with the angles of elevation from three points (\(A\), \(B\), and \(C\)) to the top of the tower. These points are collinear, meaning they lie on a straight line extending from the base of the tower. We need to find the ratio \(AB : BC\).

Step 2: Define Variables and Setup

Let's assume \(h\) is the height of the tower, \(x\) is the distance from point \(A\) to the base of the tower. Since \(B\) and \(C\) are collinear points on the same line to \(A\), let's define distances as follows: \(y\) is the distance between \(B\) and the base, and \(z\) for \(C\). Thus, \(AB = y - x\) and \(BC = z - y\).

Step 3: Apply Trigonometry

Using the definition of the tangent function, we can set up equations for each point. For angle \(30^{\circ}\) at point \(A\), we have \(\tan(30^{\circ}) = \frac{h}{x}\). For angle \(45^{\circ}\) at \(B\), \(\tan(45^{\circ}) = \frac{h}{y}\). For \(60^{\circ}\) at \(C\), \(\tan(60^{\circ}) = \frac{h}{z}\).

Step 4: Solve Each Equation

We know that \(\tan(30^{\circ}) = \frac{1}{\sqrt{3}}\), \(\tan(45^{\circ}) = 1\), and \(\tan(60^{\circ}) = \sqrt{3}\). This gives us:1. \(\frac{h}{x} = \frac{1}{\sqrt{3}} \Rightarrow x = h\sqrt{3}\)2. \(\frac{h}{y} = 1 \Rightarrow y = h\)3. \(\frac{h}{z} = \sqrt{3} \Rightarrow z = \frac{h}{\sqrt{3}}\)

Step 5: Calculate Distances and Find the Ratio

Calculate \(AB\) and \(BC\) using the values obtained:\(AB = y - x = h - h\sqrt{3}\)\(BC = z - y = \frac{h}{\sqrt{3}} - h\)Now, simplify the ratio \(AB : BC\):\(\frac{h - h\sqrt{3}}{\frac{h}{\sqrt{3}} - h}\)

Step 6: Simplify the Ratio

Upon simplification, if both terms are expressed in terms of \(h\), we can cancel the \(h\) term. The ratios become comparable as: \[ \frac{1 - \sqrt{3}}{\frac{1}{\sqrt{3}} - 1} \]After rationalizing and simplifying, this ratio simplifies to \(\sqrt{3} : 1\).

Step 7: Finalize the Answer

From the simplifications, the ratio \(AB : BC = \sqrt{3} : 1\). Thus, the correct option is (D) \(\sqrt{3} : 1\).

Key Concepts

Angle of Elevation

The **angle of elevation** is an important concept in trigonometry that refers to the angle formed between the horizontal line and the line of sight when you look up at an object. Imagine standing flat on the ground and looking up at the top of a tower or building. The angle that your line of sight makes with the horizontal ground is the angle of elevation.

Understanding this angle is crucial because it helps determine various other measurements, such as the height of the object you are looking at, especially when the distance to the object is known. This angle is measured in degrees and can easily be calculated using trigonometric functions when the other necessary values, such as height or distance, are given.

For instance, in the problem involving collinear points and a tower, the points A, B, and C offer different angles of elevation. Each provides a clue about the relative positioning and height of the tower, which can then be used for further calculations.

Tangent Function

The **tangent function** is one of the primary trigonometric functions, often represented as \( \tan \theta \). In right triangle trigonometry, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. This function is critical in solving problems involving angles of elevation.

In equations, it looks like this: \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \). In the context of the tower problem, the height of the tower is the "opposite" side, and the distance from the base tower to the observer (at points A, B, or C) is the "adjacent" side. For example, for point A where the angle of elevation is \( 30^{\circ} \), we write \( \tan(30^{\circ}) = \frac{h}{x} \), where \( h \) is the tower height and \( x \) is the distance from A to the tower base.

This function helps to set up equations for solving the unknowns like the height, which further assists in finding relationships or ratios between different points as required in the exercise.

Collinear Points

Points are said to be **collinear** if they lie on a single straight line. This concept is widely used in geometry and is significant in varied applications, including trigonometry problems.

In this exercise, points A, B, and C are collinear. In simpler terms, if you visualize a straight path or line leading to the tower’s base, these three points follow this line sequentially. It simplifies problem-solving because knowing the line is straight allows easier calculation of distances and relationships between the angles of elevation from each point to the tower.

The information about these points being collinear is essential because it helps streamline calculations and makes it easier to apply trigonometric functions since the distance from each of these points to the base of the tower can be treated linearly, aiding in the ratio calculation.

Height Calculation

**Calculating height** using trigonometry involves using known values like angles of elevation and distances from the object. With the tower exercise, these calculations become apparent when establishing equations with the tangent function.

For example, given different angles of elevation from points A, B, and C, trigonometric equations are set up as follows: for point A, \( \tan(30^{\circ}) = \frac{h}{x} \); for B, \( \tan(45^{\circ}) = \frac{h}{y} \); and for C, \( \tan(60^{\circ}) = \frac{h}{z} \). These expressions allow us to express distances x, y, and z in terms of the height h of the tower.

The ratios obtained from these calculations then help in further tasks like determining the distance between points or any other relationships required in a trigonometric problem. Thus, even if the height itself remains abstract, it provides a stepping stone to calculate further answers like the ratio of distances, which in this problem turns out to be \( \sqrt{3} : 1 \).