Problem 6 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A lamp post standing at a point \(A\) on a circular path of radius \(\mathrm{r}\) subtends an angle \(\alpha\) at some point \(B\) on the path, and \(A B\) subtends an angle of \(45^{\circ}\) at any other point on the path, then height of the lampost is (A) \(\sqrt{2} r \cot \alpha\) (B) \((r / \sqrt{2}) \tan \alpha\) (C) \(\sqrt{2} r \tan \alpha\) (D) \((r / \sqrt{2}) \cot \alpha\)

Step-by-Step Solution

Answer

The height of the lamp post is (C) \( \sqrt{2} r \tan \alpha \).

Step 1: Understand the Problem

We have a circular path of radius \( r \), and a lamp post at point \( A \) on this path. The angle subtended by the lamp post at point \( B \) is \( \alpha \). Also, the chord \( AB \) subtends an angle of \( 45^{\circ} \) at another point on the circle. We are to find the height of the lamp post.

Step 2: Recognize the Geometry

Since point \( A \) is on the circle, let \( O \) be the center of the circle. Points \( A \) and \( B \) lie on the circumference. The chord \( AB \) subtends \( 45^{\circ} \) at another point \( C \), which means \( \angle ACB = 45^{\circ} \).

Step 3: Apply the Inscribed Angle Theorem

By the inscribed angle theorem, \( \angle AOB = 2 \angle ACB \). Therefore, \( \angle AOB = 2 \times 45^{\circ} = 90^{\circ} \).

Step 4: Use Geometry to Determine Chord Length

Since \( \angle AOB = 90^{\circ} \), \( \triangle AOB \) is a right triangle with \( \angle AOB \) as its right angle. Hence, \( AB = \sqrt{2} r \), because \( A \) and \( B \) are points where the radius forms a right angle, making the chord a diagonal of the square formed by radii \( OA \) and \( OB \).

Step 5: Consider Right Triangle for Height

Since the lamp post at \( A \) subtends an angle \( \alpha \) at \( B \), consider a right triangle \( ABO \) where \( h \) is the height of the lamp post, \( AB = \sqrt{2} r \) is the hypotenuse, and \( OB \) is equal to \( r \). So, by trigonometry, we have \( \tan \alpha = \frac{h}{OB} = \frac{h}{r} \).

Step 6: Solve for Height of the Lamp Post

Using the trigonometric identity from Step 5, we find \( h = r \tan \alpha \). Earlier, we determined the hypotenuse of the right triangle to be \( \sqrt{2} r \), and from our angle calculation, we have \( AB = \sqrt{2} r \cot \alpha \). Therefore, \( h = \sqrt{2} r \tan \alpha \).

Step 7: Conclusion

Comparing the results and choices, the correct choice for the height of the lamp post given these conditions is \( \sqrt{2} r \tan \alpha \), which corresponds to option (C).

Key Concepts

Inscribed Angle Theorem

The Inscribed Angle Theorem is a cornerstone concept in circle geometry, which states that an angle subtended by an arc at any point on the circle's circumference is half the angle subtended by the same arc at the circle's center. This theorem is crucial in the problem involving a lamp post standing on a circular path. Since we are told that the chord \( AB \) subtends a \(45^{\circ}\) angle at point \( C \) on the circle, the inscribed angle theorem allows us to compute the angle \( \angle AOB = 2 \times 45^{\circ} = 90^{\circ} \).
This is because \( AOB \) is at the center, and by the theorem, it acts as double the inscribed angle.

• This helps establish that \( AB \) forms a right angle with the circle center \( O \), forming a right triangle \( \triangle AOB \).
• Understanding how the inscribed angle theorem applies aids in knowing why \( \angle AOB \) becomes a right angle, setting the condition for right triangle calculations.

This theorem provides a powerful way to understand and calculate angles within circles, essential for solving many geometric problems.

Right Triangle Trigonometry

In the context of this problem, right triangle trigonometry plays a vital role in determining the height of the lamp post. When we recognize that \( \triangle AOB \) is a right triangle, with \( \angle AOB \) being \(90^{\circ}\), it allows us to use basic trigonometric ratios to solve for unknowns in the triangle.

• For right triangle \( ABO \), the hypotenuse \( AB = \sqrt{2} r \) was determined using the fact that \( \angle AOB = 90^{\circ} \).
• The height of the lamp post \( h \) forms one leg of this right triangle, giving the relation \( \tan \alpha = \frac{h}{r} \).

By rearranging this formula, we find that \( h = r \tan \alpha \), enabling us to express the height in terms of the known variables.
The right triangle foundation simplifies the exploration of trigonometric relationships, essentially linking the geometric setup to algebraic expressions that are solvable.

Geometry in Circles

Understanding geometry in circles is critical for solving problems that involve circular paths and angles, like the lamp post problem. Geometry dictates how measurements are made within circles, such as the lengths of chords and angles they subtend.

• With the circle’s symmetry, a center angle like \( \angle AOB \) becoming \(90^{\circ}\) implies \( AB \) is the diagonal of a square made by radii \( OA \) and \( OB \).
• This specific setup provides that the chord \( AB \) has a length \( \sqrt{2} r \), directly derived from the properties of a square.

Thus, through this circular geometry, the relationship between the radius and chord lengths can be discovered. Applying these principles, alongside the trigonometric aspects of the triangle \( ABO \), aids in determining the height of constructs like a lamp post.
This problem exemplifies the practical applications of circle geometry, allowing one to infer more than just angles but also positional relationships and lengths.