Question

\(\mathrm{ABC}\) is vertical tower such that \(\mathrm{AB}=\mathrm{BC}=\mathrm{x}\). Form the foot \(\mathrm{A}\) of the tower, a horizontal line \(\mathrm{AQP}\) is drawn. If the angle of elevation of the point \(\mathrm{B}\) from \(\mathrm{P}\) is \(\alpha\) and that of \(\mathrm{C}\) from \(\mathrm{Q}\) is \(2 \alpha\), then \(\mathrm{AP}, \mathrm{x}, \mathrm{PQ}\) are in (1) A.P. (2) G.P. (3) H.P. (4) None of these

Short answer

AP, x, PQ are in G.P.

Step-by-step solution

Understand the problem setup

Visualize the vertical tower ABC where AB = BC = x. From point A, a horizontal line AQP is drawn. The angle of elevation from point P to B is α, and the angle of elevation from point Q to C is 2α.

Apply trigonometric principles

The angle of elevation from P to B (\text{\textalpha}) gives us the relationship \[ \tan(\text{\textalpha}) = \frac{x}{AP} \] Similarly, for Q to C (2\text{\textalpha}), \[ \tan(2\text{\textalpha}) = \frac{2x}{AQ} \]

Express relationships in terms of AP and AQ

From the trigonometric identities: \[ AP = \frac{x}{\tan(\text{\textalpha})} \] \[ AQ = \frac{2x}{\tan(2\text{\textalpha})} \]

Use angle of elevation properties

By trigonometric identity for double angles: \[ \tan(2\text{\textalpha}) = \frac{2 \tan(\text{\textalpha})}{1 - \tan^2(\text{\textalpha})} \]

Relate AP and AQ using tangent identity

Combine and simplify the equations using the double-angle identity: \[ AQ = \frac{2x}{\frac{2 \tan(\text{\textalpha})}{1 - \tan^2(\text{\textalpha})}} = \frac{(1 - \tan^2(\text{\textalpha}))x}{\tan(\text{\textalpha})} \] Since AP is \[ AP = \frac{x}{\tan(\text{\textalpha})} \]

Simplify AQ relationship

The equation for AQ simplifies to: \[ AQ = (1 - \tan^2(\text{\textalpha})) AP \]

Compare distances

Given AQ - AP = PQ, recognize the horizontal distance relationships determining if AP, x, PQ fit arithmetic (A.P.), geometric (G.P.), or harmonic progression (H.P.).

Conclusion

Assess the values derived. Among progression series under the original conditions, identify if it's A.P., G.P., or H.P.

Key concepts

angle of elevation

The angle of elevation is the angle formed between the horizontal line and the line of sight when looking upwards at an object. In this problem, there are two angles of elevation to consider: one from point P to B (\text{\textalpha}) and another from point Q to C (2\text{\textalpha}). The angle of elevation helps us relate the vertical heights and horizontal distances using trigonometric ratios like the tangent function. The relation using the tangent function is given by: \ \[ \tan(\text{\textalpha}) = \frac{x}{AP} \] \ \[ \tan(2\text{\textalpha}) = \frac{2x}{AQ}. \] These formulas are crucial because they let us relate AP and AQ to x, aiding in identifying the type of progression.

trigonometric identities

Trigonometric identities help simplify expressions and solve problems involving angles and distances. In this scenario, the double-angle identity for tangent is fundamental. The identity states: \ \[ \tan(2\text{\textalpha}) = \frac{2 \tan(\text{\textalpha})}{1 - \tan^2(\text{\textalpha})}. \] Using these identities, we expressed AP and AQ purely in terms of x and the angle \text{\textalpha}. This reduction simplifies our problem and provides a clearer path to find if AP, x, and PQ form a progression.

progression series

A progression series refers to a sequence of numbers with a specific relationship between consecutive terms. There are three key progression types: \

• Arithmetic Progression (A.P.): Each term is obtained by adding a constant difference to the previous term.
\
• Geometric Progression (G.P.): Each term is obtained by multiplying the previous term by a constant ratio.
\
• Harmonic Progression (H.P.): The reciprocals of the terms form an A.P.

To check the type of progression in our problem, we computed that: \ \[ AQ = \frac{(1 - \tan^2(\text{\textalpha}))x}{\tan(\text{\textalpha})} \ \text{and} \] \ \[ AP = \frac{x}{\tan(\text{\textalpha})}. \] Using the relationship AQ - AP = PQ, we can determine the type of progression by comparing the values of AP, x, and PQ.