Final answer:
The point P (m, 6) divides the line segment joining points A (-4, 3) and B (2, 8) in the ratio of -2/5, calculated using the section formula that applies to the y-coordinates. The x-coordinate 'm' can similarly be found but is not part of the provided options.
Step-by-step explanation:
The question asks to determine the ratio in which the point P (m, 6) divides the line segment joining points A (-4, 3) and B (2, 8). To solve this problem, we will use the section formula, which is used to find the coordinates of a point which divides a given line segment into a specific ratio. The formula for the point dividing AB in the ratio m:n is ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n)).
Given that P has a y-coordinate of 6, we can equate the y part of the formula to 6 and solve for the ratio. Let the ratio in which P divides AB be k:1. Then we have:
6 = (k*8 + 1*3) / (k + 1)
After cross-multiplying and solving for k, the ratio comes out to be -2/5.
To find the x-coordinate m of point P, we use the section formula again:
m = (k*2 + 1*(-4)) / (k + 1)
Substitute the value of k:
m = [(-2/5)*2 + 1*(-4)] / [(-2/5) + 1]
After simplifying this, we can find the value of m. However, from the options provided, we see that they pertain to finding the ratio, which has already been calculated as -2/5, hence the answer is a. -2/5.