The center(s) of the circle(s) passing through the points (0, 0) and (1, 0) and touching the circle `x^2+y^2=9` is (are) `(3/2,1/2)` (b) `(1/2,3/2)` `(1/2,2^(1/2))` (d) `(1/2,-2^(1/2))`

Options

A. `(3//2,1//2)`
B. `(1//2,3//2)`
C. `(1//2,2^(1//2))`
D. `(1//2,-2^(1//2))`

Explanation

To solve the problem of finding the centers of the circles that pass through the points (0, 0) and (1, 0) and touch the circle defined by the equation \( x^2 + y^2 = 9 \), we can follow these steps:

### Step-by-Step Solution

1. **Identify the Given Points and Circle**:
The points through which the circles pass are \( A(0, 0) \) and \( B(1, 0) \). The circle we are touching is defined by \( x^2 + y^2 = 9 \), which has a center at \( (0, 0) \) and a radius of \( 3 \).

2. **Determine the Midpoint of the Chord**:
The midpoint \( M \) of the chord \( AB \) can be calculated as:
\[ M = \left( \frac{0 + 1}{2}, \frac{0 + 0}{2} \right) = \left( \frac{1}{2}, 0 \right) \]

3. **Find the Perpendicular Bisector**:
The center of the circle that passes through points \( A \) and \( B \) must lie on the perpendicular bisector of the segment \( AB \). Since \( AB \) is horizontal, the perpendicular bisector is a vertical line passing through \( M \), which is \( x = \frac{1}{2} \).

4. **Determine the Distance to the Circle**:
Let the center of the desired circle be \( C\left(\frac{1}{2}, y\right) \). The distance from \( C \) to the center of the circle \( (0, 0) \) is given by:
\[ d = \sqrt{\left(\frac{1}{2} - 0\right)^2 + (y - 0)^2} = \sqrt{\frac{1}{4} + y^2} \]
For the circle to touch the circle \( x^2 + y^2 = 9 \), this distance must equal the radius of the larger circle minus the radius of the smaller circle. The radius of the larger circle is \( 3 \) and the radius of the smaller circle is \( r \) (which we need to find).

5. **Set Up the Equation**:
The distance from \( C \) to the origin must equal \( 3 - r \). The radius \( r \) of the smaller circle can be expressed as the distance from \( C \) to the line segment \( AB \) (which is \( |y| \)). Therefore, we have:
\[ \sqrt{\frac{1}{4} + y^2} = 3 - |y| \]

6. **Square Both Sides**:
Squaring both sides gives:
\[ \frac{1}{4} + y^2 = (3 - |y|)^2 \]
Expanding the right side:
\[ \frac{1}{4} + y^2 = 9 - 6|y| + y^2 \]

7. **Simplify the Equation**:
Cancel \( y^2 \) from both sides:
\[ \frac{1}{4} = 9 - 6|y| \]
Rearranging gives:
\[ 6|y| = 9 - \frac{1}{4} = \frac{36}{4} - \frac{1}{4} = \frac{35}{4} \]
Thus:
\[ |y| = \frac{35}{24} \]

8. **Find Possible Values of \( y \)**:
This gives us two possible values for \( y \):
\[ y = \frac{35}{24} \quad \text{or} \quad y = -\frac{35}{24} \]

9. **Determine the Centers**:
Therefore, the centers of the circles are:
\[ C_1\left(\frac{1}{2}, \frac{35}{24}\right) \quad \text{and} \quad C_2\left(\frac{1}{2}, -\frac{35}{24}\right) \]

### Final Answer:
The centers of the circles are \( \left(\frac{1}{2}, \frac{35}{24}\right) \) and \( \left(\frac{1}{2}, -\frac{35}{24}\right) \).