Which of the following lines lie on the plane `x+2y-z+4=0?` a. `(x-1)/1=y/(-1)=(z-5)/1` b. `x-y+z=2x+y-z=0` c. ` hat r=2 hat i- hat j+4 hat k+lambda(3 hat i+ hat j+5 hat k)` d. none of these

Options

A. `(x-1)/(1)=(y)/(-1)=(z-5)/(-1)`
B. `x-y+z=2x+y-z=0`
C. `vecr=2hati-hatj+4hatk+lamda(3hati+hatj+5hatk)`
D. none of these

Answer

a, c

Explanation

For line `(x-1)/(1)= (y)/(-1)= (z-5)/(-1)`, point `(1, 0, 5)` lies on the plane. Also, the vector alont the line `hati-hatj-hatk` is perpendicular to the normal `hati+2hatj-hatk` to the plane.
For line `vecr=2hati-hatj+4hatk+lamda(3hati+hatj+5hatk)`, point `(2, -1, 4)` lies on the plane and vectors `3hati+hatj+5hatk` is perpendicular to the normal `hati+2hatj-hatk`.
Line `x-y+z=2x+y-z=0` passes through the origin, which is not on the given plane.