If OA and OB are tangents to circle `(x-2)^2+y^2=1` from origin (O) then area of `triangle OAB` is

Options

A. `3/4`
B. `(3sqrt3)/4`
C. `(3sqrt3)/2`
D. `(sqrt3)/4`

Answer

To solve the problem, we need to find the area of triangle OAB where OA and OB are tangents to the circle given by the equation \((x-2)^2 + y^2 = 1\) from the origin O(0,0).

### Step-by-Step Solution:

1. **Identify the Circle's Center and Radius:**
The given equation of the circle is \((x-2)^2 + y^2 = 1\).
- The center of the circle \(C\) is at \((2, 0)\).
- The radius \(r\) of the circle is \(1\).

2. **Calculate the Length of the Tangents:**
The length of the tangents from a point to a circle can be calculated using the formula:
\[ L = \sqrt{d^2 - r^2} \]
where \(d\) is the distance from the point to the center of the circle, and \(r\) is the radius of the circle.

- Calculate the distance \(d\) from the origin \(O(0,0)\) to the center \(C(2,0)\):
\[ d = \sqrt{(2-0)^2 + (0-0)^2} = \sqrt{4} = 2 \]
- Now, substitute \(d\) and \(r\) into the tangent length formula:
\[ L = \sqrt{2^2 - 1^2} = \sqrt{4 - 1} = \sqrt{3} \]

3. **Determine the Coordinates of Points A and B:**
Since OA and OB are tangents to the circle, we can find the coordinates of points A and B using the tangent length and the angle formed with the x-axis. The points A and B will be at equal distances from the center along the line connecting the origin and the center of the circle.

The coordinates of points A and B can be determined using the angle \(\theta\) formed with the x-axis:
- The coordinates of point A can be expressed as:
\[ A\left(2 + \sqrt{3}\cos\theta, \sqrt{3}\sin\theta\right) \]
- The coordinates of point B can be expressed as:
\[ B\left(2 - \sqrt{3}\cos\theta, -\sqrt{3}\sin\theta\right) \]

4. **Calculate the Area of Triangle OAB:**
The area \(A\) of triangle OAB can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is the distance between points A and B, and the height is the perpendicular distance from O to line AB.

- The distance between points A and B (base):
\[ AB = 2\sqrt{3} \]
- The height from O to line AB is the radius \(r = 1\).

Therefore, the area of triangle OAB is:
\[ \text{Area} = \frac{1}{2} \times AB \times r = \frac{1}{2} \times 2\sqrt{3} \times 1 = \sqrt{3} \]

5. **Final Calculation:**
The area of triangle OAB is:
\[ \text{Area} = \sqrt{3} \]

### Conclusion:
The area of triangle OAB is \(\sqrt{3}\).