JEE Main 2024 — 9 April, Shift 1 — Question 18

$\quad \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}-\vec{b}$

$\Rightarrow(\mathrm{x}, \mathrm{y}, \mathrm{z})=(\alpha-5,1,-2)$

$\Rightarrow \mathrm{x}=\alpha-5, \mathrm{y}=1, \mathrm{z}=-2$

Area of $\Delta=5 \sqrt{6}$ (given)

$\frac{1}{2}|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}|=5 \sqrt{6}$

$$\left| \begin{matrix}i & j & k \\\alpha & 4 & 2 \\X & 1 & -2 \\\end{matrix} \right|=10\sqrt{6}$$

$\Rightarrow(2 \alpha+2 \alpha-10)^{2}+(\alpha-4 \alpha+20)^{2}=500$

$\Rightarrow(4 \alpha-10)^{2}+(20-3 \alpha)^{2}=500$

$\Rightarrow 25 \alpha^{2}-80 \alpha-120 \alpha=0$

$\Rightarrow \alpha(25 \alpha-200)=0$

$\Rightarrow \alpha=8$ (given $\alpha$ is + ve number)

$\Rightarrow \mathrm{x}=\alpha-5=3$

$|\overrightarrow{\mathrm{c}}|^{2}=\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}$

$=9+1+4$ $=14$