Question

The equation of the plane through the intersection of the plane $3x-y+2z=0$, $x+y+z-2=0$ and the point $(2,2,1)$ is

• A. $7x+5y+4z+8=0$
• B. $7x+5y+4z-8=0$
• C. $7x-5y+4z-8=0$
• D. None of these

Answer 1

The correct option is C

$7x-5y+4z-8=0$

Form the equation of the plane based on given information

Equation of the plane through the intersection of the given planes is given by

$3x-y+2z-4+\lambda (x+y+z-2)=0...\left(i\right)$ where $\lambda \in \mathrm{ℝ}$

The point $(2,2,1)$ lies on this plane. Hence it must satisfy the equation $\left(i\right)$

Substituting the co-ordinates in equation $\left(i\right)$ we get

$3\left(2\right)-2+2\left(1\right)-4+\lambda (2+2+1-2)=0$

$\Rightarrow$ $3\lambda =-2$

$\Rightarrow$ $\lambda =\frac{-2}{3}$

Re-substituting this value of $\lambda$ in equation $\left(i\right)$ we get

$\Rightarrow 3x-y+2z-4-\frac{2}{3}\left(x+y+z-2\right)=0$

$\Rightarrow$ $9x-3y+6z-12-2x-2y-2z+4=0$

$\Rightarrow$ $7x-5y+4z-8=0$

Hence, option $\left(C\right)$ $7x-5y+4z-8=0$, is the correct answer.