Question

The equation of the plane containing the line 2x - 5y + z = 3, x + y + 4z = 5 and parallel to the plane x + 3y + 6z = 1, is

• A. 2x +6y +12z =13
• B. x +3y +6z = -7
• C. x +3y +6z =7
• D. 2x +6y +12z = -13

Answer 1

The correct option is C x +3y +6z =7

Equation of the required plane is $2x-5y+z-3+\lambda (x+y+4z-5)=0\Rightarrow x(2+\lambda )+y(\lambda -5)+z(4\lambda +1)-3-5\lambda =0$

Since, this plane is parallel to $x+3y+6z=1$, we have
$\frac{\lambda +2}{1}=\frac{\lambda -5}{3}=\frac{4\lambda +1}{6}$

Solving, we get $\lambda =\frac{-11}{2}$

Substituting for $\lambda$ in the plane's equation,

$\frac{-7x}{2}-\frac{21y}{2}-21z+\frac{49}{2}=0\Rightarrow x+3y+6z-7=0$