In what ratio does the point (−4, 6) divide the line segment joining the…

To determine the ratio in which the point P(-4, 6) divides the line segment joining the points A(-6, 10) and B(3, -8), we use the section formula for internal division.

Step-by-Step Analysis
The Section Formula:
If a point P(x, y) divides the line segment joining A(x1, y1) and B(x2, y2) in the ratio m:n, the coordinates are given by:
x = (m * x2 + n * x1) / (m + n)
y = (m * y2 + n * y1) / (m + n)

Substitute the known values:
We have A(-6, 10), B(3, -8), and P(-4, 6). Let the ratio be m:n.
Using the x-coordinate:
-4 = (m * 3 + n * (-6)) / (m + n)

Solve for the ratio m:n:
-4 * (m + n) = 3m - 6n
-4m - 4n = 3m - 6n
-4m - 3m = -6n + 4n
-7m = -2n
7m = 2n
m / n = 2 / 7

The ratio m:n is 2:7.