If a chord of a circle `x ^(2) +y ^(2) =4` with one extermity at `(1, sqrt3)` subtends a right angle at the centre of this circle, then the coordinates of the other extermity of this chord can be:

Options

A. `(-1sqrt3)`
B. `(1, - sqrt3)`
C. `(-sqrt3, -1)`
D. `(sqrt3, -1)`

Answer

To solve the problem, we need to find the coordinates of the other extremity of the chord that subtends a right angle at the center of the circle defined by the equation \( x^2 + y^2 = 4 \). The circle has a radius of \( r = 2 \) (since \( r^2 = 4 \)).

### Step-by-Step Solution:

1. **Identify the center and radius of the circle:**
The equation of the circle is \( x^2 + y^2 = 4 \). The center of the circle is at \( (0, 0) \) and the radius \( r = 2 \).

2. **Given point on the chord:**
One extremity of the chord is given as \( A(1, \sqrt{3}) \).

3. **Using the property of the right angle subtended at the center:**
If a chord subtends a right angle at the center of the circle, the two points on the circle (the extremities of the chord) and the center form a right triangle. The lengths from the center to each extremity are equal to the radius.

4. **Finding the coordinates of the other extremity:**
Let the other extremity of the chord be \( B(x, y) \). Since \( A \) and \( B \) subtend a right angle at the center, the vectors \( OA \) and \( OB \) are perpendicular. This gives us the condition:
\[ OA \cdot OB = 0 \]
where \( OA = (1, \sqrt{3}) \) and \( OB = (x, y) \).

5. **Setting up the dot product:**
The dot product is given by:
\[ 1 \cdot x + \sqrt{3} \cdot y = 0 \]
Therefore, we can express \( y \) in terms of \( x \):
\[ y = -\frac{1}{\sqrt{3}} x \]

6. **Using the circle equation:**
Since point \( B \) lies on the circle, it must satisfy the circle's equation:
\[ x^2 + y^2 = 4 \]
Substituting \( y \) from the previous step:
\[ x^2 + \left(-\frac{1}{\sqrt{3}} x\right)^2 = 4 \]
Simplifying this:
\[ x^2 + \frac{1}{3} x^2 = 4 \]
\[ \frac{4}{3} x^2 = 4 \]
\[ x^2 = 3 \quad \Rightarrow \quad x = \sqrt{3} \text{ or } x = -\sqrt{3} \]

7. **Finding corresponding \( y \) values:**
For \( x = \sqrt{3} \):
\[ y = -\frac{1}{\sqrt{3}} \cdot \sqrt{3} = -1 \]
So one point is \( B(\sqrt{3}, -1) \).

For \( x = -\sqrt{3} \):
\[ y = -\frac{1}{\sqrt{3}} \cdot (-\sqrt{3}) = 1 \]
So the other point is \( B(-\sqrt{3}, 1) \).

8. **Final coordinates of the other extremity:**
The coordinates of the other extremity of the chord can be either \( (\sqrt{3}, -1) \) or \( (-\sqrt{3}, 1) \).

### Conclusion:
The coordinates of the other extremity of the chord are \( (\sqrt{3}, -1) \) and \( (-\sqrt{3}, 1) \).