if A(2,1) cuts line P(2,1) and B(8,4) then

Options

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

Explanation

To solve the problem, we need to determine the relationship between points A(2,1), P(2,1), and B(8,4) in terms of the ratio in which point P divides the line segment AB.

### Step-by-Step Solution:

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

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 can be given by:
\[ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
Here, we need to find the ratio in which P divides AB.

3. **Set Up the Equations:**
Let the ratio in which P divides AB be k:1. Thus, we can set:
\[ P\left( \frac{8k + 2}{k + 1}, \frac{4k + 1}{k + 1} \right) = (2, 1) \]

4. **Equate the x-coordinates:**
From the x-coordinates, we have:
\[ \frac{8k + 2}{k + 1} = 2 \]
Cross-multiplying gives:
\[ 8k + 2 = 2(k + 1) \]
Simplifying this, we get:
\[ 8k + 2 = 2k + 2 \]
\[ 8k - 2k = 2 - 2 \]
\[ 6k = 0 \quad \Rightarrow \quad k = 0 \]

5. **Equate the y-coordinates:**
Now, using the y-coordinates, we have:
\[ \frac{4k + 1}{k + 1} = 1 \]
Cross-multiplying gives:
\[ 4k + 1 = k + 1 \]
Simplifying this, we get:
\[ 4k - k = 1 - 1 \]
\[ 3k = 0 \quad \Rightarrow \quad k = 0 \]

6. **Interpret the Result:**
Since \( k = 0 \), this means that point P divides the line segment AB in the ratio 0:1, indicating that point P coincides with point A. Therefore, point P lies on the line segment AB but is not between A and B; it is exactly at point A.

### Conclusion:
The relationship between points A, P, and B is that point P lies at point A, and thus, it does not divide the segment AB in any meaningful ratio other than being at one of the endpoints.