The circumcentre of the triangle formed by `(0, 0), (2, -1)` and `(-1, 3)` is `(5/2, 5/2).`Then the orthocentre is

Options

A. (-4,-3)
B. (4,3)
C. (-4,3)
D. none of these

Explanation

To find the orthocenter of the triangle formed by the points \((0, 0)\), \((2, -1)\), and \((-1, 3)\) given that the circumcenter is \((\frac{5}{2}, \frac{5}{2})\), we can follow these steps:

### Step 1: Calculate the Centroid of the Triangle
The centroid \(G\) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by the formula:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
For our triangle:
- \(x_1 = 0\), \(y_1 = 0\)
- \(x_2 = 2\), \(y_2 = -1\)
- \(x_3 = -1\), \(y_3 = 3\)

Calculating the coordinates of the centroid:
\[ G_x = \frac{0 + 2 - 1}{3} = \frac{1}{3} \]
\[ G_y = \frac{0 - 1 + 3}{3} = \frac{2}{3} \]
Thus, the centroid \(G\) is \(\left(\frac{1}{3}, \frac{2}{3}\right)\).

### Step 2: Use the Relationship Between Centroid, Circumcenter, and Orthocenter
The centroid \(G\) divides the line segment joining the circumcenter \(O'\) and the orthocenter \(H\) in the ratio \(1:2\). This means:
\[ G = \left( \frac{O'_x + 2H_x}{3}, \frac{O'_y + 2H_y}{3} \right) \]
Given \(O' = \left(\frac{5}{2}, \frac{5}{2}\right)\) and \(G = \left(\frac{1}{3}, \frac{2}{3}\right)\), we can set up the equations:
\[ \frac{O'_x + 2H_x}{3} = \frac{1}{3} \]
\[ \frac{O'_y + 2H_y}{3} = \frac{2}{3} \]

### Step 3: Solve for \(H_x\) and \(H_y\)
From the first equation:
\[ O'_x + 2H_x = 1 \quad \text{(multiplying by 3)} \]
Substituting \(O'_x = \frac{5}{2}\):
\[ \frac{5}{2} + 2H_x = 1 \]
\[ 2H_x = 1 - \frac{5}{2} = \frac{2 - 5}{2} = -\frac{3}{2} \]
\[ H_x = -\frac{3}{4} \]

From the second equation:
\[ O'_y + 2H_y = 2 \quad \text{(multiplying by 3)} \]
Substituting \(O'_y = \frac{5}{2}\):
\[ \frac{5}{2} + 2H_y = 2 \]
\[ 2H_y = 2 - \frac{5}{2} = \frac{4 - 5}{2} = -\frac{1}{2} \]
\[ H_y = -\frac{1}{4} \]

### Step 4: Write the Coordinates of the Orthocenter
Thus, the coordinates of the orthocenter \(H\) are:
\[ H = \left(-\frac{3}{4}, -\frac{1}{4}\right) \]

### Final Answer
The orthocenter of the triangle is \(\left(-\frac{3}{4}, -\frac{1}{4}\right)\).