If the coordinates of orthocentre O' are centroid G of a `DeltaABC` are (0,1) and (2,3) respectively, then the coordinates of the circumcentre are

Options

A. (3,2)
B. (1,0)
C. (4,3)
D. (3,4)

Answer

To find the coordinates of the circumcenter \( C \) of triangle \( ABC \) given the coordinates of the orthocenter \( O' \) and the centroid \( G \), we can use the relationship between these points. The orthocenter \( O' \), centroid \( G \), and circumcenter \( C \) are collinear, and the centroid divides the line segment \( OC \) in the ratio \( 2:1 \).

### Step-by-Step Solution:

1. **Identify Given Points**:
- Orthocenter \( O' = (0, 1) \)
- Centroid \( G = (2, 3) \)

2. **Use the Section Formula**:
The centroid \( G \) divides the line segment \( OC \) in the ratio \( 2:1 \). If we denote the coordinates of the circumcenter \( C \) as \( (x, y) \), we can use the section formula for internal division:
\[ G = \left( \frac{2x + 0}{2 + 1}, \frac{2y + 1}{2 + 1} \right) \]

3. **Set Up the Equations**:
From the x-coordinates:
\[ \frac{2x + 0}{3} = 2 \]
From the y-coordinates:
\[ \frac{2y + 1}{3} = 3 \]

4. **Solve for \( x \)**:
Multiply both sides of the x-coordinate equation by 3:
\[ 2x = 6 \implies x = 3 \]

5. **Solve for \( y \)**:
Multiply both sides of the y-coordinate equation by 3:
\[ 2y + 1 = 9 \implies 2y = 8 \implies y = 4 \]

6. **Conclusion**:
The coordinates of the circumcenter \( C \) are \( (3, 4) \).

### Final Answer:
The coordinates of the circumcenter are \( (3, 4) \).