Find the co-ordinates of a point P on the line segment joining `A(1,2)and B(6,7)` such that `AP=2/5AB.`

Answer

To find the coordinates of point P on the line segment joining points A(1, 2) and B(6, 7) such that \( AP = \frac{2}{5} AB \), we can follow these steps:

### Step 1: Understand the relationship between AP and AB
We know that the total length of the segment AB can be expressed as:
\[ AB = AP + PB \]
Given that \( AP = \frac{2}{5} AB \), we can express PB in terms of AP:
\[ PB = AB - AP \]
Substituting for AB:
\[ PB = AP - AP = AB - \frac{2}{5} AB = \frac{3}{5} AB \]

### Step 2: Set up the ratio
From the above, we can establish the ratio of \( AP \) to \( PB \):
\[ \frac{AP}{PB} = \frac{2/5 \, AB}{3/5 \, AB} = \frac{2}{3} \]
Thus, the ratio \( AP:PB = 2:3 \).

### Step 3: Use the section formula
Using the section formula, if a point P divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, the coordinates of point P are given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Here, \( m = 2 \), \( n = 3 \), \( A(1, 2) \) and \( B(6, 7) \).

### Step 4: Calculate the x-coordinate of P
Substituting the values into the formula for x-coordinate:
\[ x = \frac{2 \cdot 6 + 3 \cdot 1}{2 + 3} = \frac{12 + 3}{5} = \frac{15}{5} = 3 \]

### Step 5: Calculate the y-coordinate of P
Now, substituting the values into the formula for y-coordinate:
\[ y = \frac{2 \cdot 7 + 3 \cdot 2}{2 + 3} = \frac{14 + 6}{5} = \frac{20}{5} = 4 \]

### Step 6: Conclusion
Thus, the coordinates of point P are:
\[ P(3, 4) \]