A horizontal plane `4x-3y+7z=0` is given. Find a line of greatest slope passes through the point `(2,1,1)` in the plane `2x+y-5z=0.`

Explanation

The required line through the point `P(2, 1, 1)` in the plane `2x+y-5z=0` and of greatest slope is perpendicular to line of intersection of the planes
`" "2x+y-5z=0 and 4x-3y+7z=0`
Vector along line of intersection of planes is
`" "|{:(hati,,hatj,,hatk),(2,,1,,-5),(4,,-3,,7):}|=-8hati-34hatj-10hatk`
Then equation of line is `(x-2)/(-8)=(y-1)/(-34)=(z-1)/(-10) or (x-2)/(4)=(y-1)/(17)=(z-1)/(5)`