JEE Main 2024 (Online) 8th April Evening ShiftQuestion 1
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 :
Points and Straight Lines
Solution
Let $u=(5,2)$, $v=(2,a)$. $\cos\frac\pi4=\frac{u\cdot v}{|u||v|}\Rightarrow 2(10+2a)^2=29(4+a^2)$. $8a^2+80a+200=29a^2+116\Rightarrow 21a^2-80a-84=0$. Roots: $a=-\frac67,\ \frac{14}{3}$; product $=-\frac{84}{21}=-4$. For both, $u\cdot v=10+2a>0$, so the angle is indeed $\frac\pi4$. Absolute product $=4$. Answer (D).