JEE Main 2022 (Online) 24th June Evening ShiftQuestion 1
Let the area of the triangle with vertices $A(1,\alpha)$, $B(\alpha,0)$ and $C(0,\alpha)$ be $4$ sq. units. If the points $(\alpha,-\alpha)$, $(-\alpha,\alpha)$ and $(\alpha^{2},\beta)$ are collinear, then $\beta$ is equal to
Points and Straight Lines
Solution
Area $=\tfrac12|1(0-\alpha)+\alpha(\alpha-\alpha)+0|=\tfrac{|\alpha|}{2}=4\Rightarrow|\alpha|=8$. $(\alpha,-\alpha)$ and $(-\alpha,\alpha)$ lie on $y=-x$, so $(\alpha^2,\beta)$ collinear gives $\beta=-\alpha^2=-64$. Answer: (C).