Straight Lines
Angle Subtended — Absolute Value of Product
nta_pyq_2024_apr
Grade 11
Question:
If the line segment joining the points $(5,2)$ and $(2,a)$ subtends an angle $\dfrac{\pi}{4}$ at the origin, then the absolute value of the product of all possible values of $a$ is:
Step-by-Step Solution
Key Concept: $m_{OA}=2/5$, $m_{OB}=a/2$. $\tan(\pi/4)=1=\left|\dfrac{m_{OA}-m_{OB}}{1+m_{OA}\cdot m_{OB}}\right|=\left|\dfrac{2/5-a/2}{1+a/5}\right|=\dfrac{|4-5a|}{|2(5+a)|}$.
$a=-6/7$ or $a=14/3$. Product $=-4$, absolute value $=4$.
Correct Answer: 4