Question 31 of 303MediumMCQ
If the line segment joining the points (5,2) and (2,a) subtends an angle 4π at the origin, then the absolute value of the product of all possible values of a is :
Solutions · 1
Official solution
To solve this problem, we will use the concept of the slope and tangent of the angle subtended by the line segments at the origin.
Given points: (5,2) and (2,a)
The slope of the line joining the origin and (5,2) is:
\[m_1 = \frac{2-0}{5-0} = \frac{2}{5}\]
The slope of the line joining the origin and (2,a) is:
\[m_2 = \frac{a-0}{2-0} = \frac{a}{2}\]
The line segment joining these points subtends an angle \[\theta = \frac{\pi}{4}\] at the origin.
According to the tangent formula for the angle between two lines:
\[ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]
Here, \[\theta = \frac{\pi}{4}\] so, \[\tan \left( \frac{\pi}{4} \right) = 1\]. Therefore,
\[ 1 = \left| \frac{\frac{a}{2} - \frac{2}{5}}{1 + \left(\frac{2}{5} \cdot \frac{a}{2}\right)} \right| \]
This simplifies to:
\[ 1 = \left| \frac{\frac{5a - 4}{10}}{1 + \frac{2a}{10}} \right| \]
Multiplying both the numerator and denominator by 10 to simplify:
\[ 1 = \left| \frac{5a - 4}{10 + 2a} \right| \]
This leads to two equations due to the absolute value:
1. 5a−4=10+2a
3a=14
\[ a = \frac{14}{3} \]
2. 5a−4=−(10+2a)
5a−4=−10−2a
7a=−6
\[ a = -\frac{6}{7} \]
Thus, the possible values of a are \[\frac{14}{3}\] and \[-\frac{6}{7}\].
The product of these values is:
\[ \left| \frac{14}{3} \times -\frac{6}{7} \right| \]
\[ = \left| -\frac{84}{21} \right| \]
\[ = \left| -4 \right| \]
=4
Therefore, the absolute value of the product of all possible values of a is 4.
Answer: Option A
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