We will use the formula:

Let R divide PQ externally in the ratio m1:m2 where P(x1, y1) and Q(x2, y2) The coordinates of R (x,y) can be found using the following formula

x = (m1x2 - m2x1)/(m1 - m2)

y = (m1y2 - m2y1)/(m1 - m2)

The point of division P (-4, 1) of the segment AB is 3/5 of the distance from one end A (2,-2) to the other end B (x, y). Find the x coordinate of B.

Case1: P divides the segment AB internally

AP: PB = 3 : 5 or we can write AP = 3t, PB = 5t and AB = AP + PB = 8t (since A,P,B are collinear)

lets observe how B(x,y) divides the segment AP:

B(x,y) divides the segment AP externally and AB:BP = 8t : 5t = 8:5 that is we can use the formula

x = (m1x2 - m2x1)/(m1 - m2)

y = (m1y2 - m2y1)/(m1 - m2)

where m1=8, m2=5, x1=2, y1 =-2, x2 =-4, y2 =1

x = (8(-4) - 5*2)/(5 - 8) => x=14

y = (8*1 - 5*(-2))/(5 - 8) => y=-6

B(14, -6)

Case2: P divides the segment AB externally and BP > AP

AP: PB = 3 : 5 or we can write AP = 3t, PB = 5t and AB = PB - AB = 5t - 3t = 2t (since A,P,B are collinear)

lets observe how B(x,y) divides the segment AP

B(x,y) divides the segment AP externally and AB:BP = 2t : 5t = 2:5 that is we can use the formula

x = (m1x2 - m2x1)/(m1 - m2)

y = (m1y2 - m2y1)/(m1 - m2)

where m1=2, m2=5, x1=2, y1 =-2, x2 =-4, y2 =1

x = (2(-4) - 5*2)/(2 - 5) => x=6

y = (2*1 - 5*(-2))/(2 - 5) => y=-4

B(6, -4)

We have 2 solutions for B: B(14, -6) and B(6, -4).