The line (x/2)=(-y/3)=(z/1) is vertical. The direction cosines of the line of greatest slope in the plane 3 x-2 y+z=5 are proportional to

Question

The line (x/2)=(-y/3)=(z/1) is vertical. The direction cosines of the line of greatest slope in the plane 3 x-2 y+z=5 are proportional to (A) (11,16,-1) (B) (-11,16,1) (C) (16,11,-1) (D) (16,11,1)

Tardigrade Admin · Accepted Answer

Equation of horizontal plane through origin

2 x - 3 y + z = 0

The direction ratios of the line of intersection of the planes

2 x - 3 y + z = 0

and

3 x - 2 y + z = 5

are given by

2 a - 3 b + c = 0

and

3 a - 2 b + c = 0

∴ \frac{a}{- 1} = \frac{b}{1} = \frac{c}{5}

Let

l, m, n

be the

D C^{'} s

of the line of greatest slope, because the line of greatest slope is perpendicular to the line of intersection of the given plane with the horizontal plane.

- l + m + 5 n = 0

and

3 l - 2 m + n = 0

So,

\frac{l}{11} = \frac{m}{16} = \frac{n}{- 1}

∴ D C^{'} s

are proportional to

(11, 16, - 1)

Q. The line $\frac{x}{2} = \frac{- y}{3} = \frac{z}{1}$ is vertical. The direction cosines of the line of greatest slope in the plane $3 x - 2 y + z = 5$ are proportional to

(11,16,-1)

(-11,16,1)

(16,11,-1)

(16,11,1)

(11,16,1)