I've seen these types of questions in OG. I try to solve them by logic, trying each point in the answer choice and make sure it satisfies the "how far" requirement.

An algebraic approach would be better.

If a Point P(x,y) divides Point A(x1, y1) and B(x2, y2) in the ratio of a:b, then the co-ordinates of point P are:
x = (a*x2 + b*x1) / (a + b)
y = (a*y2 + b*y1) / (a + b)

(Notice that it's a cross ratio)

In the problem stated by you, if A(x1, y1) = (-5, 6) and B(x2, y2) = (2, 0) and the required Point P(x,y) divides A and B in a:b = 3:1

Use the formula above, and get the answer.

rajman41 wrote:Q: What are the coordinates for the point on line A(-5,6) and (-2,0) that is 3 times as far from b, and that is in between points A and B?

could you please help me break down in a more understandable process to solve this problem in a short time period. Thank you in advance!

I have rewritten the question stem. There is something missing in it.

What are the co-ordinates for the point on line A(-5,6) and B(-2,0) that is 3 times as far from B as from A, and that is in between points A and B?

Say The point 3 times farther from B is C. So A, C & B are co-linear & lying on the line in that orer. We can say AC:BC::1:3

Say the co-ordinates of C is (x,y).

Co-ordinates of a point that intersects a line in the ratio of m:n is given by

x = (mx1+nx2)/(m+n) & y = (my1+ny2)/(m+n); here (x1,y1)=(-5,6) i.e. point A & (x2,y2)=(-2,0) i.e. point B.

x = (3.-5+1.-2)/(3+1) = -32/4 = -8 & y = (3.6+1.0)/(3+1) = 18/4 = 9/2. So the point is (-8,9/2).