Mathematics - Vector Question with Solution | TestHub

$\begin{array}{rl}& L_1:\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\\ & L_1:\frac{x}{1}=\frac{y}{\alpha }=\frac{z-5}{1}\\ & \vec{x}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 3& 4\\ 1& \alpha & 1\end{array}\right|=\hat{\mathrm{i}}(3-4\alpha )-\hat{\mathrm{j}}(-2)+\hat{\mathrm{k}}(2\alpha -3)\\ & \text{ S.D. }=\left|\frac{\overrightarrow{\mathrm{BA}}\cdot \overrightarrow{\mathrm{n}}}{|\overrightarrow{\mathrm{n}}|}\right|=\left|\frac{(\hat{\mathrm{i}}+2\hat{\mathrm{j}}-2\hat{\mathrm{k}})\cdot \vec{n}}{|\overrightarrow{\mathrm{n}}|}\right|\\ & \Rightarrow 6(13-8\alpha {)}^2=25\left((4\alpha -3{)}^2+(2\alpha -3{)}^2+16\right)\\ & 6\left(64a^2-280\alpha +169\right)=25\left(20{\alpha }^2-36\alpha +34\right)\\ & \Rightarrow 116{\alpha }^2+348\alpha -164=0\\ & {\alpha }_1+{\alpha }_2=\frac{-348}{116}=-3\end{array}$