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 find the ratio in which the point P(2,1) divides the line segment joining points A(4,2) and B(8,4), we can use the section formula. Here’s a step-by-step solution:

### Step 1: Understand the Section Formula
The section formula states that if a point P(x, y) divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of P can be given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

### Step 2: Assign Coordinates
Let:
- A = (4, 2) → (x1, y1)
- B = (8, 4) → (x2, y2)
- P = (2, 1) → (x, y)

### Step 3: Set Up the Ratio
Assume point P divides the line segment AB in the ratio \( m:n \). We can express the coordinates of P using the section formula:
\[ P\left(\frac{8m + 4n}{m+n}, \frac{4m + 2n}{m+n}\right) = (2, 1) \]

### Step 4: Set Up the Equations
From the x-coordinates:
\[ \frac{8m + 4n}{m+n} = 2 \]
From the y-coordinates:
\[ \frac{4m + 2n}{m+n} = 1 \]

### Step 5: Solve the First Equation
Cross-multiply the first equation:
\[ 8m + 4n = 2(m + n) \]
Expanding gives:
\[ 8m + 4n = 2m + 2n \]
Rearranging:
\[ 8m - 2m + 4n - 2n = 0 \implies 6m + 2n = 0 \implies 3m + n = 0 \implies n = -3m \]

### Step 6: Solve the Second Equation
Now, substitute \( n = -3m \) into the second equation:
\[ \frac{4m + 2(-3m)}{m + (-3m)} = 1 \]
This simplifies to:
\[ \frac{4m - 6m}{m - 3m} = 1 \implies \frac{-2m}{-2m} = 1 \]
This is always true, confirming our ratio.

### Step 7: Find the Ratio
From \( n = -3m \), we can express the ratio as:
\[ m:n = m:-3m = 1:-3 \]
Thus, the ratio in which P divides the line segment AB is \( -1:3 \).

### Step 8: Conclusion
Since the ratio is negative, it indicates that point P lies outside the segment AB, specifically extending beyond point A.