To find the ratio in which the x-axis divides the line segment joining the points P(5, 3) and Q(2, -6), we can follow these steps:
### Step 1: Understand the problem
We need to find the point on the x-axis (which has coordinates of the form (x, 0)) that divides the line segment joining points P(5, 3) and Q(2, -6) in a certain ratio.
### Step 2: Set up the ratio
Let the ratio in which the x-axis divides the line segment be k:1. This means we can denote the point of intersection on the x-axis as S(x, 0).
### Step 3: Use the section formula
According to the section formula, the coordinates of the point S that divides the line segment joining points (x1, y1) and (x2, y2) in the ratio m:n are given by:
\[
S\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
Here, we have:
- P(5, 3) → (x1, y1) = (5, 3)
- Q(2, -6) → (x2, y2) = (2, -6)
- Ratio = k:1 → m = k, n = 1
### Step 4: Write the y-coordinate equation
Since the point S lies on the x-axis, its y-coordinate is 0. Therefore, we can set up the equation for the y-coordinate:
\[
\frac{k \cdot (-6) + 1 \cdot 3}{k + 1} = 0
\]
This simplifies to:
\[
\frac{-6k + 3}{k + 1} = 0
\]
### Step 5: Solve for k
To find k, we set the numerator equal to zero:
\[
-6k + 3 = 0
\]
Solving for k gives:
\[
-6k = -3 \quad \Rightarrow \quad k = \frac{3}{6} = \frac{1}{2}
\]
### Step 6: State the ratio
The ratio in which the x-axis divides the line segment joining points P and Q is therefore:
\[
1:2
\]
### Final Answer
Thus, the required ratio is **1:2**.
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