If the point P (2, 1) lies on the line segment joining points A(4, 2) and B(8, 4), then

Options

A. `AP = (1)/(3) AB`
B. AP = PB
C. `PB = (1)/(3) AB`
D. `AP = (1)/(2) AB`

Answer

To solve the problem, we need to determine the relationship between the point P(2, 1) and the line segment joining points A(4, 2) and B(8, 4). We will use the section formula to find the ratio in which point P divides the line segment AB.

### Step-by-Step Solution:

1. **Identify the Coordinates:**
- Let A = (4, 2)
- Let B = (8, 4)
- Let P = (2, 1)

2. **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:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

3. **Set Up the Equations:**
We can set the coordinates of P equal to the coordinates derived from the section formula:
- For the x-coordinate:
\[ 2 = \frac{8m + 4n}{m+n} \]
- For the y-coordinate:
\[ 1 = \frac{4m + 2n}{m+n} \]

4. **Cross-Multiply to Eliminate the Denominator:**
- For the x-coordinate:
\[ 2(m+n) = 8m + 4n \implies 2m + 2n = 8m + 4n \implies 6m = 2n \implies n = 3m \]
- For the y-coordinate:
\[ 1(m+n) = 4m + 2n \implies m + n = 4m + 2n \implies 3m = n \]

5. **Substitute n in Terms of m:**
From the equation \(n = 3m\), we can substitute this into the ratio:
\[ \frac{AP}{PB} = \frac{m}{n} = \frac{m}{3m} = \frac{1}{3} \]

6. **Conclusion:**
The point P divides the line segment AB in the ratio 1:3 externally.

### Final Answer:
The ratio in which point P divides the line segment AB is \(AP:PB = 1:3\).

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