The ratio in which x-axis divides the line segment joining the point (2, 3) and (4, - 8) is

Answer

To find the ratio in which the x-axis divides the line segment joining the points (2, 3) and (4, -8), we can 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 m:n, then the coordinates of point P are given by:

\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

In this case, we want to find the ratio in which the x-axis (where y = 0) divides the line segment.

### Step 1: Identify the points
Let A = (2, 3) and B = (4, -8).

### Step 2: Set up the section formula
Let the ratio in which the x-axis divides the line segment be k:1. Therefore, we can denote m = k and n = 1.

### Step 3: Apply the section formula for y-coordinate
Using the section formula for the y-coordinate, we have:

\[ y = \frac{ky_2 + 1y_1}{k + 1} \]

Substituting the coordinates of points A and B:

\[ 0 = \frac{k(-8) + 1(3)}{k + 1} \]

### Step 4: Solve for k
Setting the equation to zero gives:

\[ 0 = \frac{-8k + 3}{k + 1} \]

This implies that the numerator must be zero:

\[ -8k + 3 = 0 \]

Solving for k:

\[ 8k = 3 \implies k = \frac{3}{8} \]

### Step 5: Write the ratio
The ratio in which the x-axis divides the line segment is k:1, which is:

\[ \frac{3}{8}:1 \implies 3:8 \]

Thus, the x-axis divides the line segment joining the points (2, 3) and (4, -8) in the ratio **3:8**.

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