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JEE Main 2016

JEE Main 2016 Mathematics question on Three Dimensional Geometry

If the line, x−32=y+2−1=z+43\dfrac{x-3}{2} = \dfrac{y+2}{-1} = \dfrac{z+4}{3} lies in the plane, lx+my−z=9lx + my - z = 9, then l2+m2l^2 + m^2 is equal to
  1. A.
    26
  2. B.
    18
  3. C.
    5
  4. D.
    2
Show answer and solution

Correct option: D

Solution

The line passes through (3, -2, -4) and has direction (2, -1, 3).

Since the line lies in the plane lx + my - z = 9:

1) The point (3, -2, -4) satisfies the plane: 3l - 2m + 4 = 9 → 3l - 2m = 5.

2) The direction vector is perpendicular to the plane's normal (l, m, -1): 2l - m - 3 = 0 → 2l - m = 3, so m = 2l - 3.

Substituting into 3l - 2(2l-3) = 5: 3l - 4l + 6 = 5 → l = 1, m = -1.

Therefore l² + m² = 1 + 1 = 2.

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