QJEEM22Ma1568: JEE Main 2022 Mathematics - Class 11 JEEM22Ma1568 Solution: Let the area of the triangle with vertices A(1, ), B( , 0) and…

$\because$ A(1, $\alpha$ ), B( $\alpha$ , 0) and C(0, $\alpha$ ) are the vertices of $\Delta$ ABC and area of $\Delta$ ABC = 4

$\therefore$ $$\left|\frac{1}{2}\left|\begin{array}{ccc}1& \alpha & 1\\ \alpha & 0& 1\\ 0& \alpha & 1\end{array}\right|\right|=4$$

$\Rightarrow \left| {1(1 - \alpha ) - \alpha (\alpha ) + {\alpha ^2}} \right| = 8$

$\Rightarrow \alpha = \, \pm \,8$

Now, $(\alpha ,\, - \alpha ),\,( - \alpha ,\alpha )$ and $({\alpha ^2},\beta )$ are collinear

$\therefore$ $$\left|\begin{array}{ccc}8& -8& 1\\ -8& 8& 1\\ 64& \beta & 1\end{array}\right|=0=\left|\begin{array}{ccc}-8& 8& 1\\ 8& -8& 1\\ 64& \beta & 1\end{array}\right|$$

$\Rightarrow 8(8 - \beta ) + 8( - 8 - 64) + 1( - 8\beta - 8 \times 64) = 0$

$\Rightarrow 8 - \beta - 72 - \beta - 64 = 0$

$\Rightarrow \beta = - 64$