Question

The equation of the plane that contains the point $A$ $\left(1,-1,2\right)$ and is perpendicular to each of the planes $2x+3y-2z=5$ and $x+2y-3z=8$ is

• A. $5x+4y-z=7$
• B. $5x-4y+z=7$
• C. $-5x+4y-z=7$
• D. $5x-4y-z=7$

Answer 1

The correct option is D

$5x-4y-z=7$

Form the equation of the plane based on given information

The equation of a plane passing through $\left(1,-1,2\right)$ is given as

$a(x-1)+b(y+1)+c(z-2)=0...\left(i\right)$

The required the lane is also perpendicular to the planes $2x+3y-2z=5$ and $x+2y-3z=8$ respectively

Hence the direction ratios of the plane follow the relations

$2a+3b-2c=0$

$a+2b-3c=0$

Solving these two equations by cross multiplication

$\frac{a}{-5}=\frac{b}{4}=\frac{c}{1}$

Substitute $a=-5c,b=4c$ in equation $\left(i\right)$ we get

$-5c(x-1)+4c(y+1)+c(z-2)=0$

$\Rightarrow$ $-5x+5+4y+4+z-2=0$

$\Rightarrow$ $5x-4y-z=7$

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