Circle C (not shown) is drawn on a coordinate plane, centered at the origins. If the point (a, b) lies on the circumference of the circle, what is the radius of the cirle in terms of a and b?

Options

A. `a-b`
B. `a+b`
C. `sqrt(a^(2)+b^(2))`
D. `a^(2)+b^(2)`

Answer

To find the radius of circle C centered at the origin (0, 0) in terms of the coordinates of a point (a, b) on its circumference, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Center of the Circle**:
The center of the circle is given as the origin, which has coordinates (0, 0).

2. **Identify the Point on the Circumference**:
The point on the circumference of the circle is given as (a, b).

3. **Use the Distance Formula**:
The radius of the circle is the distance between the center and the point on the circumference. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane is calculated using the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

4. **Substitute the Coordinates into the Distance Formula**:
Here, \((x_1, y_1) = (0, 0)\) (the center) and \((x_2, y_2) = (a, b)\) (the point on the circumference). Plugging these values into the distance formula gives:
\[ d = \sqrt{(a - 0)^2 + (b - 0)^2} \]

5. **Simplify the Expression**:
This simplifies to:
\[ d = \sqrt{a^2 + b^2} \]

6. **Conclusion**:
Therefore, the radius \(r\) of the circle in terms of \(a\) and \(b\) is:
\[ r = \sqrt{a^2 + b^2} \]

### Final Answer:
The radius of the circle in terms of \(a\) and \(b\) is \(r = \sqrt{a^2 + b^2}\).

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