MCQ
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$2$
- ✓$4$
✓
Answer
Correct option: D.
$4$
d
$Image$
$Image$
$ \mathrm{m}_{\mathrm{OA}}=\frac{2}{5} $
$ \mathrm{~m}_{\mathrm{OB}}=\frac{\mathrm{a}}{2}$
$\tan \frac{\pi}{4}=\left|\frac{2}{5}-\frac{a}{2}\right|$
$1=\left|\frac{4-5 a}{10+2 a}\right|$
$4-5 \mathrm{a}= \pm(10+2 \mathrm{a}) $
$4-5 \mathrm{a}=10+2 \mathrm{a} $ $ 4-5 \mathrm{a}=-10-2 \mathrm{a} $
$\Rightarrow 7 \mathrm{a}+6=0 $ $ 3 \mathrm{a}=14 $
$\Rightarrow \mathrm{a}=-\frac{6}{7} $ $ \mathrm{a}=+\frac{14}{3}$
$-\frac{6}{7} \times \frac{14}{3}=-4$
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