If the line segment joining the points (5, 2) and (2, a) subtends an angle at the origin, then the absolute value of the product of all possible values of a is: [2024]

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Solution

Q.
If the line segment joining the points (5, 2) and (2, a) subtends an angle $\frac{π}{4}$ at the origin, then the absolute value of the product of all possible values of a is: [2024]

1 8

2 2

3 4

4 6

Ans.
(3)

$m_{1} = \frac{2}{5}, m_{2} = \frac{a}{2}$

Angle between the lines is $π / 4$

Since, $\tan θ =$ $| \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} |$

$\Rightarrow$ $\tan \frac{π}{4} =$ $| \frac{\frac{2}{5} - \frac{a}{2}}{1 + \frac{2}{5} \times \frac{a}{2}} |$ $\Rightarrow$ $| \frac{\frac{2}{5} - \frac{a}{2}}{1 + \frac{a}{5}} |$ $= 1$

$\Rightarrow$ $| \frac{4 - 5 a}{10 + 2 a} |$ $= 1 \Rightarrow \frac{4 - 5 a}{10 + 2 a} = \pm 1$

$\Rightarrow \frac{4 - 5 a}{10 + 2 a} = 1 o r \frac{4 - 5 a}{10 + 2 a} = - 1$

$\Rightarrow 4 - 5 a = 10 + 2 a or 4 - 5 a = - 10 - 2 a$

$\Rightarrow a = - \frac{6}{7} or a = \frac{14}{3}$

$∴$ Required product $=$ $| - \frac{6}{7} \times \frac{14}{3} |$ $= 4$ .

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