A circle passing through (0,0),(2,6),(6,2) cuts the x-axis at the point `Pne(0,0)`. Then the length of OP, where O is the origin, is

Options

A. `5/2`
B. `5/sqrt2`
C. 5
D. 10

Answer

To solve the problem, we need to find the length of OP, where O is the origin (0,0) and P is the point where the circle intersects the x-axis, given that the circle passes through the points (0,0), (2,6), and (6,2).

### Step-by-Step Solution:

1. **Equation of the Circle**:
The general equation of a circle can be expressed as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
Since the circle passes through the origin (0,0), substituting \(x = 0\) and \(y = 0\) gives:
\[ c = 0 \]
Therefore, the equation simplifies to:
\[ x^2 + y^2 + 2gx + 2fy = 0 \]

2. **Substituting the Second Point (2,6)**:
Substitute \(x = 2\) and \(y = 6\) into the equation:
\[ 2^2 + 6^2 + 2g(2) + 2f(6) = 0 \]
Simplifying this gives:
\[ 4 + 36 + 4g + 12f = 0 \implies 4g + 12f = -40 \quad \text{(Equation 1)} \]

3. **Substituting the Third Point (6,2)**:
Substitute \(x = 6\) and \(y = 2\) into the equation:
\[ 6^2 + 2^2 + 2g(6) + 2f(2) = 0 \]
Simplifying this gives:
\[ 36 + 4 + 12g + 4f = 0 \implies 12g + 4f = -40 \quad \text{(Equation 2)} \]

4. **Solving the System of Equations**:
We now have two equations:
\[ 4g + 12f = -40 \quad \text{(1)} \]
\[ 12g + 4f = -40 \quad \text{(2)} \]
To eliminate \(g\), we can multiply Equation (1) by 3:
\[ 12g + 36f = -120 \quad \text{(3)} \]
Now subtract Equation (2) from Equation (3):
\[ (12g + 36f) - (12g + 4f) = -120 + 40 \]
This simplifies to:
\[ 32f = -80 \implies f = -\frac{5}{2} \]

5. **Finding g**:
Substitute \(f = -\frac{5}{2}\) back into Equation (1):
\[ 4g + 12(-\frac{5}{2}) = -40 \]
Simplifying gives:
\[ 4g - 30 = -40 \implies 4g = -10 \implies g = -\frac{5}{2} \]

6. **Equation of the Circle**:
Now substituting \(g\) and \(f\) back into the circle equation:
\[ x^2 + y^2 - 5x - 5y = 0 \]

7. **Finding Intersection with the x-axis**:
To find where the circle intersects the x-axis, set \(y = 0\):
\[ x^2 - 5x = 0 \]
Factoring gives:
\[ x(x - 5) = 0 \]
Thus, \(x = 0\) or \(x = 5\). Since point P cannot be the origin, we have:
\[ P = (5, 0) \]

8. **Calculating Length OP**:
The length of OP (distance from origin O to point P) is given by:
\[ OP = \sqrt{(5 - 0)^2 + (0 - 0)^2} = \sqrt{25} = 5 \]

### Final Answer:
The length of OP is \(5\).