Find the coordinates of the point which divides the line joining (5, -2) and (9, 9) in the ratio `3 : 1`.

Answer

To find the coordinates of the point that divides the line segment joining the points (5, -2) and (9, 9) in the ratio 3:1, 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:

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

### Step 1: Identify the coordinates and the ratio
Let:
- A = (5, -2) → (x1, y1) = (5, -2)
- B = (9, 9) → (x2, y2) = (9, 9)
- The ratio m:n = 3:1, so m = 3 and n = 1.

### Step 2: Calculate the x-coordinate
Using the formula for the x-coordinate:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
Substituting the values:
\[ x = \frac{3 \cdot 9 + 1 \cdot 5}{3 + 1} = \frac{27 + 5}{4} = \frac{32}{4} = 8 \]

### Step 3: Calculate the y-coordinate
Using the formula for the y-coordinate:
\[ y = \frac{my_2 + ny_1}{m+n} \]
Substituting the values:
\[ y = \frac{3 \cdot 9 + 1 \cdot (-2)}{3 + 1} = \frac{27 - 2}{4} = \frac{25}{4} = 6.25 \]

### Step 4: Write the coordinates of point P
Thus, the coordinates of the point P that divides the line segment in the ratio 3:1 are:
\[ P\left(8, 6.25\right) \]

### Final Answer
Therefore, the required point is \( (8, 6.25) \).

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