If the line, $\frac{\mathrm{x}-3}{2}=\frac{\mathrm{y}+2}{-1}=\frac{\mathrm{z}+4}{3}$ lies in the plane, $l \mathrm{x}+\mathrm{my}-\mathrm{z}=9$, then $l^{2}+\mathrm{m}^{2}$ is equal to:

JEE Main · 2016 · 03 Apr 2016 · Offline · Three Dimensional Geometry

If the line,

\frac{x - 3}{2} = \frac{y + 2}{- 1} = \frac{z + 4}{3}

lies in the plane,

l x + my - z = 9

, then

l^{2} + m^{2}

is equal to:

(A) $18$
(B) $5$
(C) $2$ ✓ Correct
(D) $26$

Answer: (C)

Solution

The line has direction

(2, - 1, 3)

and passes through

(3, - 2, - 4)

. Lying in the plane

l x + m y - z = 9

needs two conditions. First, the direction is perpendicular to the normal

(l, m, - 1)

2 l - m - 3 = 0

. Second, the point lies in the plane:

3 l - 2 m + 4 = 9

, i.e.

3 l - 2 m = 5

. From the first,

m = 2 l - 3

; substituting,

3 l - 2 (2 l - 3) = 5

gives

- l + 6 = 5

, so

l = 1

and

m = - 1

. Then

l^{2} + m^{2} = 1 + 1 = 2

JEE Main · 2016 · 03 Apr 2016 · Offline · Single correct option

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