Question

If a plane passing through the point $(2,2,1)$ and is perpendicular to the planes $3x+2y+4z+1=0$ and $2x+y+3z+2=0$. Then, the equation of the plane is

• A. $2x-y-z-1=0$
• B. $2x+3y+z-1=0$
• C. $2x+y+z+3=0$
• D. $x-y+z-1=0$

Answer 1

The correct option is A $2x-y-z-1=0$

Equation of plane passing through $(2,2,1)$ is

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

Since, above plane is perpendicular to

$3x+2y+4z+1=0$

and $2x+y+3z+2=0$

$\therefore 3a+2b+4c=0....\left(ii\right)$

and $2a+b+3c=0....\left(iii\right)$

$[\because$ for perpendicular, $a_1{a}_2+b_1{b}_2+c_1{c}_2=0]$

On multiplying eq. (iii) by $2$, we get

$4a+2b+6c=0....\left(iv\right)$

On subtracting eq, (iv) from eq. (ii), we get

$⟹c=\frac{-a}{2}$

On putting $c=\frac{-a}{2}$ in eq. (iii), we get $b=\frac{-a}{2}$

On putting $b=\frac{-a}{2}$ and $c=\frac{-a}{2}$ in eq. (i),

we get $a(x-2)-\frac{a}{2}(y-2)-\frac{a}{2}(z-1)=0$

$⟹\frac{a}{2}\left[2\right(x-2)-(y-2)-(z-1\left)\right]=0$

$⟹2x-4-y+2-z+1=0$

$⟹2x-y-z-1=0$ is the required equation of plane.