The plane containing the line and also containing its projection on the plane , contains which one of the following points?
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: C
Solution
The required plane contains the given line and its projection on 2x+3y-z=5, so it contains the given line and is parallel to the normal of 2x+3y-z=5 (since the projection lies in that plane, the plane through the line and its projection must contain the normal direction (2,3,-1)).
The required plane passes through point (3,-2,1) on the line, has direction vector of line d=(2,-1,3), and also contains direction n=(2,3,-1) (normal to the given plane).
Normal to required plane = d × n = |i j k; 2 -1 3; 2 3 -1|
= i((-1)(-1)-(3)(3)) - j((2)(-1)-(3)(2)) + k((2)(3)-(-1)(2))
= i(1-9) - j(-2-6) + k(6+2)
= (-8, 8, 8), simplify to (-1,1,1).
Equation: -1(x-3)+1(y+2)+1(z-1)=0 → -x+3+y+2+z-1=0 → -x+y+z+4=0 → x-y-z=4.
Check (2,0,-2): 2-0-(-2)=4 ✓
The required plane passes through point (3,-2,1) on the line, has direction vector of line d=(2,-1,3), and also contains direction n=(2,3,-1) (normal to the given plane).
Normal to required plane = d × n = |i j k; 2 -1 3; 2 3 -1|
= i((-1)(-1)-(3)(3)) - j((2)(-1)-(3)(2)) + k((2)(3)-(-1)(2))
= i(1-9) - j(-2-6) + k(6+2)
= (-8, 8, 8), simplify to (-1,1,1).
Equation: -1(x-3)+1(y+2)+1(z-1)=0 → -x+3+y+2+z-1=0 → -x+y+z+4=0 → x-y-z=4.
Check (2,0,-2): 2-0-(-2)=4 ✓
[JEE Main 2019 · 11 Jan · Shift I]