A point P on the x-axis divides the line segment joining the points (4, 5) and (1, -3) in certain ratio . Find the co-ordinates of point P.

Answer

To find the coordinates of point P on the x-axis that divides the line segment joining the points (4, 5) and (1, -3) in a certain ratio, we can follow these steps:

### Step 1: Identify the points and the coordinates of point P
We have two points:
- Point A (4, 5)
- Point B (1, -3)

Point P lies on the x-axis, which means its y-coordinate is 0. We can denote the coordinates of point P as (x, 0).

### Step 2: Assume the ratio in which point P divides the line segment
Let point P divide the line segment AB in the ratio k:1.

### Step 3: Apply 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 our case:
- \(x_1 = 4\), \(y_1 = 5\)
- \(x_2 = 1\), \(y_2 = -3\)
- The ratio is \(k:1\)

### Step 4: Calculate the y-coordinate of point P
Since point P lies on the x-axis, its y-coordinate is 0. Therefore, we can set up the equation:
\[ \frac{k \cdot (-3) + 1 \cdot 5}{k + 1} = 0 \]

### Step 5: Solve for k
Multiplying both sides by \(k + 1\) gives:
\[ k \cdot (-3) + 5 = 0 \]
\[ -3k + 5 = 0 \]
\[ 3k = 5 \]
\[ k = \frac{5}{3} \]

### Step 6: Calculate the x-coordinate of point P
Now, we can find the x-coordinate of point P using the section formula:
\[ x = \frac{k \cdot x_2 + 1 \cdot x_1}{k + 1} \]
Substituting the values we have:
\[ x = \frac{\frac{5}{3} \cdot 1 + 1 \cdot 4}{\frac{5}{3} + 1} \]
\[ = \frac{\frac{5}{3} + 4}{\frac{5}{3} + \frac{3}{3}} = \frac{\frac{5}{3} + \frac{12}{3}}{\frac{8}{3}} = \frac{\frac{17}{3}}{\frac{8}{3}} = \frac{17}{8} \]

### Step 7: Write the coordinates of point P
Thus, the coordinates of point P are:
\[ P\left(\frac{17}{8}, 0\right) \]

### Step 8: Find the ratio of division
The ratio in which point P divides the line segment can be expressed as:
\[ k:1 = \frac{5}{3}:1 = 5:3 \]

### Final Answer
The coordinates of point P are \(\left(\frac{17}{8}, 0\right)\) and it divides the line segment in the ratio \(5:3\).

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