Find the ratio in which point P(2,1) divides the line joining the points A (4,2) and B(8,4)

Answer

To find the ratio in which the point P(2,1) divides the line joining the points A(4,2) and B(8,4), we will 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 calculated using the following formulas:

\[ P_x = \frac{mx_2 + nx_1}{m+n} \]
\[ P_y = \frac{my_2 + ny_1}{m+n} \]

### Step 1: Assign the coordinates
Let the coordinates of A be \( A(4, 2) \) and B be \( B(8, 4) \). The coordinates of point P are \( P(2, 1) \).

### Step 2: Assume the ratio
Let the ratio in which P divides AB be \( k:1 \). Therefore, we can set \( m = k \) and \( n = 1 \).

### Step 3: Write the equations using the section formula
Using the section formula for the x-coordinate:
\[ 2 = \frac{k \cdot 8 + 1 \cdot 4}{k + 1} \]
Using the section formula for the y-coordinate:
\[ 1 = \frac{k \cdot 4 + 1 \cdot 2}{k + 1} \]

### Step 4: Solve the y-coordinate equation
From the y-coordinate equation:
\[ 1 = \frac{4k + 2}{k + 1} \]
Cross-multiplying gives:
\[ 1(k + 1) = 4k + 2 \]
This simplifies to:
\[ k + 1 = 4k + 2 \]
Rearranging gives:
\[ 1 - 2 = 4k - k \]
\[ -1 = 3k \]
Thus,
\[ k = -\frac{1}{3} \]

### Step 5: Interpret the result
The negative value of \( k \) indicates that point P divides the line segment externally. The ratio of AP to PB is given by the absolute value of \( k \), which is \( \frac{1}{3} \).

### Conclusion
Thus, point P divides the line segment AB in the ratio \( 1:3 \) externally.

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