Short Answer 
The answer outlines the steps to find a point P that divides line segment AB, defined by endpoints A(-3, 9) and B(4, 2), in a specific ratio. It details the use of formulas for calculating coordinates based on the ratio and demonstrates how to derive the final division ratio of m:n = 4:3.
Step 1: Define Points
Identify the coordinates of points A and B which are the endpoints of the line segment. This is essential as it forms the basis of your calculation. The points are as follows:
- Point A has coordinates (-3, 9).
- Point B has coordinates (4, 2).
Step 2: Use a Parameter for Division
Let point P(x, y) be the point on line segment AB that divides it in the ratio of m:n. Use the following formulas to calculate the coordinates of P based on the division ratio:
- x can be expressed as: x = (m * xB + n * xA) / (m + n).
- y can be expressed as: y = (m * yB + n * yA) / (m + n).
Step 3: Substitute and Solve for Ratio
Substitute the values of y from the line equation into the calculated expressions for y. This allows you to set up an equation that can be simplified:
- Equate the derived expression for y with y = 2 + 3x.
- Simplify the equation to isolate m and n: 16n = 12m.
- Thus, the final ratio of division is m:n = 4:3.