Let area of triangle with vertices A (1, α) , B (α, 0) and C (0, α) be 4, sq. units. If point (α

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3 and 4 .Determinants and Matrices

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Let the area of the triangle with vertices $A (1, \alpha)$, $B (\alpha, 0)$ and $C (0, \alpha)$ be $4\, sq.$ units. If the point $(\alpha,-\alpha),(-\alpha, \alpha)$ and $\left(\alpha^{2}, \beta\right)$ are collinear, then $\beta$ is equal to

A

$64$

B

$-8$

C

$-64$

D

$512$

(JEE MAIN-2022)

Solution

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

$\alpha=\pm 8$

Now given points $(8,-8),(-8,8),(64, \beta)$

$OR (-8,8),(8,-8),(64, \beta)$

are collinear $\Rightarrow$ Slope $=-1$.

$\beta=-64$

Standard 12

Mathematics

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