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
• Correct answer
  D

  -4

Solution

$\begin{aligned} & \mathrm{m}_{\mathrm{OA}}=\frac{2}{5} \\ & \mathrm{~m}_{\mathrm{OB}}=\frac{\mathrm{a}}{2}\end{aligned}$ $\begin{aligned} & \tan \frac{\pi}{4}=\left|\frac{2}{5}-\frac{a}{2}\right| \\ & 1=\left|\frac{4-5 a}{10+2 a}\right|\end{aligned}$

Hence, the correct answer is option: 'D'.