ABC is a Delta such A,B,C are at a distance 5 units from the origin. If `((1)/(3)-(1)/(3))` is the centroid of `DeltaABC` then the orthocentre is

Options

A. `(1,-1)`
B. `(-2,2)`
C. `(-1,1)`
D. `(2,-2)`

Explanation

To find the orthocenter of triangle ABC given that the centroid is at \((\frac{1}{3}, -\frac{1}{3})\) and the points A, B, and C are each at a distance of 5 units from the origin, we can follow these steps:

### Step 1: Identify the circumcenter and circumradius
Since points A, B, and C are at a distance of 5 units from the origin, the circumradius \(R\) is 5. The circumcenter \(P\) of the triangle is at the origin, which is \((0, 0)\).

**Hint:** The circumcenter is the center of the circle that passes through all vertices of the triangle.

### Step 2: Use the relationship between the orthocenter, centroid, and circumcenter
The relationship between the orthocenter \(H\), centroid \(G\), and circumcenter \(P\) is given by the formula:
\[ H = 3G - 2P \]

### Step 3: Substitute the known values
We know that:
- \(G = \left(\frac{1}{3}, -\frac{1}{3}\right)\)
- \(P = (0, 0)\)

Substituting these values into the formula:
\[ H = 3\left(\frac{1}{3}, -\frac{1}{3}\right) - 2(0, 0) \]

### Step 4: Calculate the orthocenter
Calculating the right-hand side:
\[ H = \left(3 \cdot \frac{1}{3}, 3 \cdot -\frac{1}{3}\right) - (0, 0) \]
\[ H = \left(1, -1\right) - (0, 0) \]
\[ H = (1, -1) \]

### Conclusion
The orthocenter \(H\) of triangle ABC is at the coordinates \((1, -1)\).

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