If the line segment joining the points (5, 2) and (2, a) subtends an angle π / 4 at the origin, then the absolute value

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 : 

(1) 6 

(2) 8 

(3) 2 

(4) 4 

Answer 1

Correct option is :  (4) 4 

 

\(M_{OA} = \frac{2}{5}\) 

\(M_{OB} = \frac{a}{2}\) 

\(=\tan \frac{\pi}{4}=\left|\frac{\frac{2}{5}-\frac{a}{2}}{1+\frac{2}{5} \cdot \frac{a}{2}}\right|\) 

\( \Rightarrow 1=\left|\frac{4-5 \mathrm{a}}{10+2 \mathrm{a}}\right| \Rightarrow \mathrm{a}=\frac{-6}{7}, \frac{14}{3}\) 

Product \(\Rightarrow-4\), Abs. value \(=4\)