The plane passing through the point (4, -1, 2) and parallel to the lines (x+2)/3 = (y-2)/(-1) = (z+1)/2 and (x-2)/1 = (y-3)/2 = (z-4)/3 also passes through the point

The plane passing through the point (4, -1, 2) and parallel… — JEE Main 2019 Mathematics PYQ

JEE Main 2019 · 10 Jan · Shift I

The plane passing through the point

(4, -1, 2)

and parallel to the lines

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

and

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

also passes through the point

A.
$(−1, −1, −1)$
B.
$(1, 1, −1)$
C.
$(1, 1, 1)$
D.
$(−1, 1, 1)$

Show answer and solution

Correct option: C

Solution

The normal to the plane is the cross product of the two direction vectors (3, -1, 2) and (1, 2, 3).

n = (3,-1,2) × (1,2,3) = ((-1)(3)-(2)(2), (2)(1)-(3)(3), (3)(2)-(-1)(1)) = (-3-4, 2-9, 6+1) = (-7, -7, 7)

Simplifying: normal direction is (1, 1, -1).

Plane through (4,-1,2): 1(x-4) + 1(y+1) - 1(z-2) = 0 → x + y - z - 1 = 0.

Check point (1,1,1): 1 + 1 - 1 - 1 = 0 ✓

[JEE Main 2019 · 10 Jan · Shift I]

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