If the centrroid and circumentre of a triangle are (3,3) and (6,2) respectively, then the orthocentre, is

Options

A. (-3,5)
B. (-3,1)
C. (3,-1)
D. (9,5)

Explanation

To find the orthocenter of a triangle given the coordinates of the centroid and circumcenter, we can use the relationship that the centroid divides the line segment joining the orthocenter (H) and circumcenter (O) in the ratio 2:1.

### Step-by-Step Solution:

1. **Identify the Given Points:**
- Centroid (G) = (3, 3)
- Circumcenter (O) = (6, 2)
- Let the coordinates of the orthocenter (H) be (x, y).

2. **Use the Section Formula:**
The centroid divides the line segment joining the orthocenter and circumcenter in the ratio 2:1. According to the section formula, if a point G divides the line segment joining points (x1, y1) and (x2, y2) in the ratio m:n, then:
\[ G_x = \frac{mx_2 + nx_1}{m+n} \]
\[ G_y = \frac{my_2 + ny_1}{m+n} \]

3. **Set Up the Equations:**
Here, m = 2, n = 1, (x1, y1) = (x, y) (orthocenter), and (x2, y2) = (6, 2) (circumcenter).
- For x-coordinate:
\[ 3 = \frac{2 \cdot 6 + 1 \cdot x}{2 + 1} \]
- For y-coordinate:
\[ 3 = \frac{2 \cdot 2 + 1 \cdot y}{2 + 1} \]

4. **Solve for x:**
From the x-coordinate equation:
\[ 3 = \frac{12 + x}{3} \]
Multiply both sides by 3:
\[ 9 = 12 + x \]
Rearranging gives:
\[ x = 9 - 12 = -3 \]

5. **Solve for y:**
From the y-coordinate equation:
\[ 3 = \frac{4 + y}{3} \]
Multiply both sides by 3:
\[ 9 = 4 + y \]
Rearranging gives:
\[ y = 9 - 4 = 5 \]

6. **Conclusion:**
The coordinates of the orthocenter (H) are (-3, 5).

### Final Answer:
The orthocenter of the triangle is at the point (-3, 5).