Let the point P (4,m) divide the line segment joining A (2,3) and B (6,-3) in the ratio of k:1

Using section formula

{(mx2+nx1)/(m+n), (my2+ny1)/(m+n)},

=} (4,m) = {(6k+2)/(k+1), (-3k+3)/(k+1)}

So 4 = (6k+2)/(k+1).

Solving we get k= 1.

So the ratio is 1:1.
i.e., P is the mid point of AB.

Also m = (-3k+3)/(k+1)
=} (-3+3)/1+1 = 0.

So m=0

That's all🙂

Ratio in which P(4,m) divides the line segment AB:
To find the ratio in which P(4,m) divides the line segment joining points A(2,3) and B(6,-3), we can use the concept of section formula. The section formula states that if a point P divides a line segment AB in the ratio m:n, then the coordinates of P can be found using the formula:

P(x,y) = (n * A + m * B) / (m + n)

Here, A and B are the coordinates of points A and B respectively, and m and n are the ratios in which P divides the line segment AB.

Finding the coordinates of point P:
Given that P divides the line segment joining A(2,3) and B(6,-3), we can substitute the given values into the section formula:

P(x,y) = (n * A + m * B) / (m + n)
P(x,y) = (n * (2,3) + m * (6,-3)) / (m + n)
P(x,y) = ((2n, 3n) + (6m, -3m)) / (m + n)
P(x,y) = (2n + 6m, 3n - 3m) / (m + n)

Since P(x,y) = (4, m), we can equate the x and y coordinates:

2n + 6m = 4 ...(1)
3n - 3m = m ...(2)

Solving the equations:
We now have two equations with two variables. We can solve these equations to find the values of m and n.

From equation (2), we can rearrange to isolate n:
3n - 3m = m
3n = 4m
n = (4/3)m

Substituting this value of n into equation (1):
2(4/3)m + 6m = 4
(8/3)m + 6m = 4
(8m + 18m) / 3 = 4
26m = 12
m = 12/26
m = 6/13

Substituting the value of m back into equation (2) to find n:
3n - 3(6/13) = 6/13
3n - 18/13 = 6/13
3n = 24/13
n = 8/13

Conclusion:
Therefore, the ratio in which P(4,m) divides the line segment joining A(2,3) and B(6,-3) is 6:8 or 3:4. The value of m is 6/13.