To solve the problem of finding the ratio in which the line segment joining points A(2, -3) and B(5, 6) is divided by the x-axis, and to find the coordinates of the point of division, we will follow these steps:
### Step 1: Identify the coordinates of points A and B
Given:
- A = (2, -3)
- B = (5, 6)
### Step 2: Determine the coordinates of the point of division on the x-axis
Since the point of division lies on the x-axis, its y-coordinate will be 0. Let's denote the point of division as P(x, 0).
### Step 3: Use the section formula
The section formula states that if a point P divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio k:1, then the coordinates of point P are given by:
\[
P\left(\frac{kx_2 + x_1}{k + 1}, \frac{ky_2 + y_1}{k + 1}\right)
\]
Here, we know that the y-coordinate of P is 0. Thus, we can set up the equation for the y-coordinate:
\[
\frac{ky_2 + y_1}{k + 1} = 0
\]
Substituting the coordinates of A and B:
\[
\frac{k(6) + (-3)}{k + 1} = 0
\]
### Step 4: Solve for k
To solve for k, we can set the numerator equal to zero:
\[
6k - 3 = 0
\]
Solving this gives:
\[
6k = 3 \implies k = \frac{3}{6} = \frac{1}{2}
\]
### Step 5: Determine the ratio
The ratio in which the line segment is divided is k:1, which is:
\[
\frac{1}{2}:1 \implies 1:2
\]
### Step 6: Find the x-coordinate of point P
Now, we will find the x-coordinate of point P using the section formula:
\[
x = \frac{kx_2 + x_1}{k + 1}
\]
Substituting the values:
\[
x = \frac{\left(\frac{1}{2}\right)(5) + 2}{\frac{1}{2} + 1}
\]
Calculating the numerator:
\[
= \frac{\frac{5}{2} + 2}{\frac{3}{2}} = \frac{\frac{5}{2} + \frac{4}{2}}{\frac{3}{2}} = \frac{\frac{9}{2}}{\frac{3}{2}} = \frac{9}{3} = 3
\]
### Step 7: Coordinates of point P
Thus, the coordinates of point P are:
\[
P(3, 0)
\]
### Final Answer
The line segment joining A(2, -3) and B(5, 6) is divided by the x-axis in the ratio 1:2, and the coordinates of the point of division are (3, 0).
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