Question
Under which one of the following conditions will the two planes $$x + y + z = 7$$ and $$\alpha x + \beta y + \gamma z = 3,$$ be parallel (but not coincident) ?
A.
$$\alpha = \beta = \gamma = 1\,\,\left( {{\text{only}}} \right)$$
B.
$$\alpha = \beta = \gamma = \frac{3}{7}\,\,\left( {{\text{only}}} \right)$$
C.
$$\alpha = \beta = \gamma $$
D.
none of the above
Answer :
$$\alpha = \beta = \gamma $$
Solution :
Given equation of planes are :
$$x + y + z = 7$$ and $$\alpha x + \beta y + \gamma z = 3$$
For these planes to be parallel, coefficients of $$x,\,y$$ and $$z$$ should be same i.e.
$$ \Rightarrow \alpha = \beta = \gamma $$