Mathematics - Vector Question with Solution | TestHub

$\begin{array}{rl}& \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}\\ & \Rightarrow (\mathrm{x},\mathrm{y},\mathrm{z})=(\alpha -5,1,-2)\\ & \Rightarrow \mathrm{x}=\alpha -5,\mathrm{y}=1,\mathrm{z}=-\mathrm{2.....}(1)\end{array}$
Area of $\Delta =5\sqrt{6}$ (given) $\begin{array}{rl}& \frac{1}{2}|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{c}}|=5\sqrt{6}\\ & \Vert \begin{array}{ccc}\hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}}\\ \alpha & 4& 2\\ \mathrm{x}& 1& -2\end{array}\Vert =10\sqrt{6}\end{array}$ $\begin{array}{rl}& \Rightarrow |-10\hat{\mathrm{i}}-\hat{\mathrm{j}}(-2\alpha -2\mathrm{x})+\hat{\mathrm{k}}(\alpha -4\mathrm{x})|=10\sqrt{6}\\ & \Rightarrow (2\alpha +2\alpha -10{)}^2+(\alpha -4\alpha +20{)}^2=500\\ & \Rightarrow (4\alpha -10{)}^2+(20-3\alpha {)}^2=500\\ & \Rightarrow 25{\alpha }^2-80\alpha -120\alpha =0\\ & \Rightarrow \alpha (25\alpha -200)=0\\ & \Rightarrow \alpha =8\text{ (given }\alpha \text{ is +ve number) }\\ & \Rightarrow \mathrm{x}=\alpha -5=3\\ & |\overrightarrow{\mathrm{c}}{|}^2={\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2\\ & =9+1+4\\ & =14\end{array}$