Find the point which divides the line segment joining
(-1,2), (4,-5) in the ratio 3 : 2

Answer

To find the point that divides the line segment joining the points (-1, 2) and (4, -5) in the ratio 3:2, we can 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 as follows:

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

### Step 1: Identify the coordinates and the ratio
Let:
- A = (-1, 2) → (x1, y1)
- B = (4, -5) → (x2, y2)
- m = 3 (the part of the ratio towards point B)
- n = 2 (the part of the ratio towards point A)

### Step 2: Substitute the values into the formula
Using the section formula:

\[ P\left(\frac{3 \cdot 4 + 2 \cdot (-1)}{3 + 2}, \frac{3 \cdot (-5) + 2 \cdot 2}{3 + 2}\right) \]

### Step 3: Calculate the x-coordinate
Calculating the x-coordinate:

\[ x = \frac{3 \cdot 4 + 2 \cdot (-1)}{3 + 2} = \frac{12 - 2}{5} = \frac{10}{5} = 2 \]

### Step 4: Calculate the y-coordinate
Calculating the y-coordinate:

\[ y = \frac{3 \cdot (-5) + 2 \cdot 2}{3 + 2} = \frac{-15 + 4}{5} = \frac{-11}{5} \]

### Step 5: Combine the results
Thus, the coordinates of the point P that divides the line segment in the ratio 3:2 are:

\[ P\left(2, -\frac{11}{5}\right) \]

### Final Answer
The point that divides the line segment joining (-1, 2) and (4, -5) in the ratio 3:2 is \( P\left(2, -\frac{11}{5}\right) \).

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