Question

A plane passing through the point $(3,1,1)$ contains two lines whose direction ratios are $(1,-2,2)$ and $(2,3,-1)$ respectively. If this plane also passes through the point $(\alpha ,-3,5)$, then $\alpha$ is equal to

• A. $-5$
• B. $10$
• C. $5$
• D. $-10$

Answer 1

The correct option is C

$5$

Explanation for the correct option:

Step 1. normal vector to the plane is

$\overrightarrow{n}=\left(\hat{i}-2\hat{j}+2\hat{k}\right)\times \left(2\hat{i}+3\hat{j}-\hat{k}\right)$

$\begin{array}{rcl}\overrightarrow{n}& =& \left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -2& 2\\ 2& 3& -1\end{array}\right|\\ & =& \hat{i}\left(2-6\right)-\hat{j}(-1-4)+\hat{k}(3+4)\\ & =& -4\hat{i}+5\hat{j}+7\hat{k}\end{array}$

Step 2. equation of plane passing through $\left(3,1,1\right)$ is

$\Rightarrow$ $-4\left(x-3\right)+5(y-1)+7(z-1)=0$

$\Rightarrow$$-4x+12+5y-5+7z-7=0$

$\Rightarrow$ $-4x+5y+7z=\hspace{0.17em}0$ ……(1)

Step 3. Plane is also passing through(α, –3, 5), then

$\Rightarrow$$–4\alpha +5\times (–3)+7\times \left(5\right)=0$

$\Rightarrow$ $–4\alpha –15+35=0$

$\Rightarrow$ $\mathit{\alpha }\mathbf{}\mathbf{=}\mathbf{}\mathbf{5}$

Hence, Option ‘C’ is Correct.