Sign In

JEE MAIN - Mathematics (2023 - 29th January Morning Shift - No. 27)

Let the co-ordinates of one vertex of $$\Delta ABC$$ be $$A(0,2,\alpha)$$ and the other two vertices lie on the line $${{x + \alpha } \over 5} = {{y - 1} \over 2} = {{z + 4} \over 3}$$. For $$\alpha \in \mathbb{Z}$$, if the area of $$\Delta ABC$$ is 21 sq. units and the line segment $$BC$$ has length $$2\sqrt{21}$$ units, then $$\alpha^2$$ is equal to ___________.
Answer
9

Explanation

A. $\left(\mathrm{O}_{1} 2, \alpha\right)$

JEE Main 2023 (Online) 29th January Morning Shift Mathematics - 3D Geometry Question 110 English Explanation

$\left|\frac{1}{2} \cdot 2 \sqrt{21}.\right| \begin{array}{ccc}\mathrm{i} & \mathrm{j} & \mathrm{k} \\ \alpha & 1 & \alpha+4 \\ 5 & 2 & 3\end{array}\left|\frac{1}{\sqrt{25+4+9}}\right|=21 \sqrt{21}$

$\sqrt{(2 \alpha+5)^{2}+(2 \alpha+20)^{2}+(2 \alpha-5)^{2}}=\sqrt{21} \sqrt{38}$

$\Rightarrow 12 \alpha^{2}+80 \alpha+450=798$

$\Rightarrow 12 \alpha^{2}+80 \alpha-348=0$

$\Rightarrow \alpha=3 \Rightarrow \alpha^{2}=9$

Comments (0)

Advertisement