Question

Find the equation of the plane passing through the intersection of the planes $3x-y+2z-4=0$ and $x+y+z+2=0$ and the point $(2,2,1)$.

Answer 1

Given $P_1:3x-y+2z-4=0$

$P_2:x+y+z+2=0$

Family of lines passing through same line

Family of planes is represented by $P_1+\lambda P_2$

$3x-y+2z-4+\lambda (x+y+z+2)=0$

Above plane passes through $(2,2,1)$

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

$\lambda =\frac{-2}{7}$

$P_3:3x-y+2z-\frac{2}{7}(x+y+z+2)=0$

$P_3:19x-9y+12z-32=0$