Question
The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes $$2x+y-2z=5$$ and $$3x-6y-2z=7,$$ is :
A.
$$14x+2y-15z=1$$
B.
$$14x-2y+15z=27$$
C.
$$14x+2y+15z=31$$
D.
$$-14x+2y+15z=3$$
Answer :
$$14x+2y+15z=31$$
Solution :
The required equation of plane is given by
\[\begin{array}{l}
\left| \begin{array}{l}
x - 1\,\,\,\,\,y - 1\,\,\,\,\,z - 1\\
\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\, - 2\\
\,\,\,\,\,3\,\,\,\,\,\,\,\,\,\,\, - 6\,\,\,\,\,\,\,\, - 2
\end{array} \right| = 0\\
\Rightarrow \left( {x - 1} \right)\left( { - 14} \right) - \left( {y - 1} \right)\left( 2 \right) + \left( {z - 1} \right)\left( { - 15} \right) = 0\\
\Rightarrow 14x - 14 + 2y - 2 + 15z - 15 = 0\\
\Rightarrow 14x + 2y + 15z = 31
\end{array}\]