To find the ratio in which point 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 allows us to find the coordinates of a point that divides a line segment in a given ratio.

Section Formula:
The section formula states that if a line segment joining two points A(x1, y1) and B(x2, y2) is divided by a point P(x, y) in the ratio m:n, then the coordinates of P are given by:
x = (mx2 + nx1)/(m + n)
y = (my2 + ny1)/(m + n)

Now, let's apply the section formula to solve the problem.

Given:
Point A(x1, y1) = A(2, 3)
Point B(x2, y2) = B(6, -3)
Point P(x, y) = P(4, m)

Finding the ratio:
We need to find the ratio in which point P divides the line segment AB. Let's assume the ratio m:n. Since the coordinates of P are P(4, m), we can use the section formula to find the values of x and y.

x = (mx2 + nx1)/(m + n)
4 = (m * 6 + n * 2)/(m + n) ----(1)

y = (my2 + ny1)/(m + n)
m = (m * -3 + n * 3)/(m + n) ----(2)

To solve these equations, we can use the elimination method by multiplying equation (2) by (m + n) to eliminate the denominators.

4(m + n) = (m * 6 + n * 2) ----(3)
m(m + n) = (m * -3 + n * 3) ----(4)

Expanding equation (3):
4m + 4n = 6m + 2n

Expanding equation (4):
m^2 + mn = -3m + 3n

Simplifying equation (3):
2n = 2m

Substituting the value of n in terms of m in equation (4):
m^2 + m(2m) = -3m + 3(2m)
m^2 + 2m^2 = -3m + 6m
3m^2 = 3m
m^2 - m = 0
m(m - 1) = 0

So, m = 0 or m = 1.

Case 1: m = 0
If m = 0, then n = 0 since n = 2m. However, the ratio m:n cannot be 0:0 as it would imply dividing by zero, which is undefined. Therefore, m = 0 is not a valid solution.

Case 2: m = 1
If m = 1, then n = 2m = 2(1) = 2. The ratio m:n is 1:2.

Therefore, point P(4, m) divides the line segment joining points A(2, 3) and B(6