Question

If the line segment joining the points \((5, 2)\) and \((2, a)\) subtends an angle \(\frac{\pi}{4}\) at the origin, then the absolute value of the product of all possible values of \(a\) is:

• A. 6
• B. 8
• C. 2
• D. 4

Answer 1

The Correct Option is D

Consider the points \( A(5, 2) \) and \( B(2, a) \) joined by line segments from the origin \( O \). The given condition states that these segments subtend an angle \( \frac{\pi}{4} \) at the origin. Let the slopes of the lines \( OA \) and \( OB \) be \( M_{OA} \) and \( M_{OB} \) respectively.

The slope of line \( OA \) is: \[ M_{OA} = \frac{2}{5}. \] 

The slope of line \( OB \) is: \[ M_{OB} = \frac{a}{2}. \] 

Since the angle between the lines is \( \frac{\pi}{4} \), 

we use the formula: \[ \tan \left( \frac{\pi}{4} \right) = \left| \frac{M_{OB} - M_{OA}}{1 + M_{OA} \cdot M_{OB}} \right|. \] 

Given \( \tan \left( \frac{\pi}{4} \right) = 1 \), 

we have: \[ 1 = \left| \frac{\frac{a}{2} - \frac{2}{5}}{1 + \frac{2}{5} \cdot \frac{a}{2}} \right|. \] 

Simplifying the expression: \[ \left| \frac{5a - 4}{10 + 2a} \right| = 1. \] 

Clearing the fractions: \[ \left| \frac{5a - 4}{10 + 2a} \right| = 1. \] 

This gives two cases: \[ \frac{5a - 4}{10 + 2a} = 1 \quad \text{or} \quad \frac{5a - 4}{10 + 2a} = -1. \] 

Case 1: \[ \frac{5a - 4}{10 + 2a} = 1. \] 

Cross-multiplying: \[ 5a - 4 = 10 + 2a. \] 

Rearranging terms: \[ 3a = 14 \implies a = \frac{14}{3}. \] 

Case 2: \[ \frac{5a - 4}{10 + 2a} = -1. \] 

Cross-multiplying: \[ 5a - 4 = -10 - 2a. \] 

Rearranging terms: \[ 7a = -6 \implies a = -\frac{6}{7}. \] 

The product of all possible values of \( a \) is: \[ a_1 \times a_2 = \left( \frac{14}{3} \right) \times \left( -\frac{6}{7} \right) = -4. \] 

The absolute value of the product is: \[ |a_1 \times a_2| = 4. \] 

Therefore: \[ 4. \]