Question

On a coordinates plane, a line passes through ( – 2 , 3 ) and ( 2 , 6 ). Which of the followings lies on the same line.
a. ( – 2 , 12 )
b. ( 6 , 12 )
c. ( – 6 , 0 )
d. ( – 6 , 6 )

1. By direct comparison and definition of line segment  we notice that the point (x, y) = (- 6, 0) lies on the line segment AB as each AP is a multiple of former. (Correct choice: C)

### What point lies on a line segment?

According to linear algebra, a point lies in a line segment if its vector is a multiple of the vector that generates the line segment itself, that is:
AB = k · AP        (1)
The vector that generates the line segment is:
AB = (2, 6) – (- 2, 3)
AB = (4, 3)
And the vectors related to each point are:
Case A
AP = (- 2, 12) – (- 2, 3)
AP = (0, 9)
Case B
AP = (6, 12) – (- 2, 3)
AP = (8, 9)
Case C
AP = (- 6, 0) – (- 2, 3)
AP = (- 4, – 3)
Case D
AP = (- 6, 6) – (- 2, 3)
AP = (- 4, 3)
By direct comparison and definition of line segment  we notice that the point (x, y) = (- 6, 0) lies on the line segment AB as each AP is a multiple of former. (Correct choice: C)