The coordinates of the point O which divides the join of `A(5, -2) and B(9, 6)` in the ratio `3:1` are

Options

A. `(6,0)`
B. `(9,5)`
C. `(8,4)`
D. `(4,8)`

Answer

To find the coordinates of the point O that divides the line segment joining points A(5, -2) and B(9, 6) in the ratio 3:1, we can use the section formula. The section formula for a point dividing a line segment internally in the ratio \( m:n \) is given by:

\[ \left( \frac{m x_1 + n x_2}{m+n}, \frac{m y_1 + n y_2}{m+n} \right) \]

where \( (x_1, y_1) \) are the coordinates of point A, and \( (x_2, y_2) \) are the coordinates of point B.

### Step 1: Identify the coordinates and the ratio
- Let \( A(5, -2) \) be \( (x_1, y_1) \)
- Let \( B(9, 6) \) be \( (x_2, y_2) \)
- The ratio is \( m:n = 3:1 \)

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

\[ x = \frac{3 \cdot 5 + 1 \cdot 9}{3 + 1} \]
\[ y = \frac{3 \cdot (-2) + 1 \cdot 6}{3 + 1} \]

### Step 3: Calculate the x-coordinate
Now, calculate the x-coordinate:

\[ x = \frac{15 + 9}{4} = \frac{24}{4} = 6 \]

### Step 4: Calculate the y-coordinate
Now, calculate the y-coordinate:

\[ y = \frac{-6 + 6}{4} = \frac{0}{4} = 0 \]

### Step 5: Combine the results
Thus, the coordinates of point O are:

\[ O(6, 0) \]

### Final Answer
The coordinates of the point O which divides the join of A and B in the ratio 3:1 are \( (6, 0) \).

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