Let the plane pass through the point (1,0,1) and perpendicular to the planes 2 x+3 y-z=2 and x-y+2 z=1 . Let the equation of the plane passing through the point (11,7,5) and parallel to the plane be a x+b y-z-d=0 . Then, a b + b d =

Step-by-step solution

The equation of plane passes through the point (1,0,1) is a(x-1)+b(y-0)+c(z-1)=0 ...(i) If this plane is perpendicular to the planes 2 x+3 y-z=2 and x-y+2 z=1 Then, 2 a+3 b-c=2 and a-b+2 c=1 aligned & a 6-1 = b -1-4 = c 2(-1)-3 & a 5 = b -3 = c -5 =k aligned Let a=5 k, b=-3 k, c=-5 k Putting the values of a, b and c in Eq. (i), we get the required equation of plane 5 k(x-1)+(-3 k)(y)+(-5 k)(z-1)=0 aligned & 5 x-5-3 y-5 z+5=0 & 5 x-3 y-5 z=0 aligned Now, plane passing through (11,7,5) and parallel to the plane (5 x-