The segment from A (2, 3) to B(0, 1) passes through the point C so that AC=4AB. Find the coordinates of C.

this means that C divides the segment AB externally since AC > AB and

Case1: B is between A and C: AC : CB = 4AB : (AC - AB) = 4AB : (4AB - AB) = 4 : 3

Case2: A is between C and B: AC : CB = 4AB : (AC + AB) = 4AB : (4AB + AB) = 4 : 5

We will use the following 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)

Case1: AC : CB = 4:3 => m1 = 4, m2 = 3, x1=2, y1=3, x2=0, y2=1

x = (4*0 - 3*2)/(4 - 3) = -6

y = (4*1 - 3*3)/(4 - 3) = -5

C(-6, -5)

Case1: AC : CB = 4:5 => m1 = 4, m2 = 5, x1=2, y1=3, x2=0, y2=1

x = (4*0 - 5*2)/(4 - 5) = 10

y = (4*1 - 5*3)/(4 - 5) = 11

C(10, 11)

There are 2 solutions for C: C(-6, -5) and C(10, 11).