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 :
Correct Option: 1
Solution:
$\because$ Plane contains two lines
$\therefore \vec{n}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & 2 \\ 2 & 3 & -1\end{array}\right|$
$=i(2-6)-j(-1-4)+k(3+4)=-4 i+5 j+7 k$
So, equation of plane is
$-4(x-3)+5(y-1)+7(z-1)=0$
$\Rightarrow-4 x+12+5 y-5+7 z-7=0$
$\Rightarrow-4 x+5 y+7 z=0$
This also passes through $(\alpha,-3,5)$
So, $-4 \alpha-15+35=0$
$\Rightarrow-4 \alpha=-20 \Rightarrow \alpha=5$