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

Answer: (D)

4

Answer 2

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: $4.$