Find the ratio in which P(4,m) divides the line segment joining the points A(2,3) and B(6, -3). Hence find m.

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 section formula. The section formula states that if a point divides a line segment in the ratio k:1, then the coordinates of the point can be calculated as follows:

Section Formula

If point P divides the line segment AB in the ratio k:1, then:

• x-coordinate of P = (kx_B + x_A) / (k + 1)
• y-coordinate of P = (ky_B + y_A) / (k + 1)

Given Points

Let’s denote:

• A(2, 3)
• B(6, -3)
• P(4, m)

Finding the Ratio

We need to find the ratio k such that:

• 4 = (k * 6 + 2) / (k + 1)
• m = (k * (-3) + 3) / (k + 1)

Solving for k

Starting with the x-coordinate:

4(k + 1) = 6k + 2

4k + 4 = 6k + 2

2 = 2k

Thus, k = 1.

Finding m

Now substituting k back into the equation for y:

m = (1 * (-3) + 3) / (1 + 1)

m = (-3 + 3) / 2 = 0 / 2 = 0.

Final Result

The point P(4, m) divides the line segment in the ratio 1:1, and the value of m is 0.