If the image of the point in the plane measured parallel to the line is , then is equal to
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: B
Solution
The line through P(1,-2,3) parallel to direction (1,4,5) is:
x = 1+t, y = -2+4t, z = 3+5t
Substitute into 2x+3y-4z+22=0:
2(1+t) + 3(-2+4t) - 4(3+5t) + 22 = 0
2+2t - 6+12t - 12-20t + 22 = 0
6 - 6t = 0 → t = 1
So the line meets the plane at M = (2, 2, 8).
Since M is the midpoint of PQ (image measured parallel to line means M is midpoint):
Q = (2·2-1, 2·2-(-2), 2·8-3) = (3, 6, 13)
PQ = distance from P(1,-2,3) to Q(3,6,13):
PQ = √((3-1)² + (6+2)² + (13-3)²) = √(4+64+100) = √168 = 2√42
x = 1+t, y = -2+4t, z = 3+5t
Substitute into 2x+3y-4z+22=0:
2(1+t) + 3(-2+4t) - 4(3+5t) + 22 = 0
2+2t - 6+12t - 12-20t + 22 = 0
6 - 6t = 0 → t = 1
So the line meets the plane at M = (2, 2, 8).
Since M is the midpoint of PQ (image measured parallel to line means M is midpoint):
Q = (2·2-1, 2·2-(-2), 2·8-3) = (3, 6, 13)
PQ = distance from P(1,-2,3) to Q(3,6,13):
PQ = √((3-1)² + (6+2)² + (13-3)²) = √(4+64+100) = √168 = 2√42
[JEE Main 2017]