Question

Points P and Q trisect the line segment joining the points A(−2, 0) and B(0, 8) such that P is near to A. Find the coordinates of points P and Q.

Answer 1

P and Q trisect line joining the points A and B. 
Let the coordinates of P and Q be (a, b) and (c, d) respectively. 
P is the midpoint of AQ.

`(-2 +"c")/2 = "a" and (0+"d")/2 = "b"`

⇒ c= 2a + 2 and d = 2b   ...........(1)

Also, Q is the mid point of PB.

`("a"+0)/2 ="c" and ("b"+8)/2 = "d"`

⇒ `"a"/2 = "c" and ("b"+8)/2 ="d"`   .........(2)

From (1) and (2) we have

2a + 2 =`"a"/2`

⇒ 4a + 4 = a

⇒ 3a = -4

⇒`"a" = -4/3`

Also,

2b `=("b"+8)/2`

⇒ 4b = b +8

⇒ b = `8/3`

Putting these values of a and b in (2)

`((-4)/3)/2 = "c"`

⇒ `(-2)/3 = "c"`

And 

`(8/3+8)/2 = "d"`

`((8+24)/3)/2 = "d"`

⇒`(32/2)/2 = "d"`

⇒ `16/3 = "d"`

Thus, the points are P `((-4)/3 , 8/3)` and Q `((-2)/3 , 16/3).`