The equation of the plane passing through the line of intersection of the planes x+y+z=1,2 x+3 y+4 z=7 , and perpendicular to the plane x-5 y+3 z=5 is given by

Step-by-step solution

Given planes, p₁: x+y+z=1 p ₂: 2 x +3 y +4 z =7 So, equation of plane passing through intersection of planes p₁ and p₂ is aligned & x+y+z-1+k(2 x+3 y+4 z-7)=0 & x+y+z-1+2 k x+3 k y+4 k z-7 k=0 & x(1+2 k)+y(1+3 k)+z(1+4 k)-1-7 k=0 aligned This is perpendicular to x-5 y+3 z=5 . aligned & x -5 y +3 z -5=0 & 1(1+2 k )-5(1+3 k )+3(1+4 k )=0 & 1+2 k -5-15 k +3+12 k =0 & - k -1=0 k =-1 aligned Equation of plane is aligned & x+y+z-1-1(2 x+3 y+4 z-7)=0 & x+y+z-1-2 x-3 y-4 z+7=0 & -x-2 y-3 z+6=0 & x+2 y+3 z-6=0 aligned