Geometry

Step 1: Find the midpoint of the diameter.
The center of the circle lies at the midpoint of the diameter. The midpoint of the line segment joining the points and is calculated using the midpoint formula:
Substituting the coordinates of the points: Thus, the center of the circle is .

Step 2: Calculate the radius of the circle.
The radius is the distance from the center of the circle to either endpoint of the diameter. We can use the distance formula to find this distance. Using the point as one endpoint: Substituting and : Thus, the radius is 3.

Step 3: Write the equation of the circle.
The general equation of a circle with center and radius is: Substituting , , and :
Step 4: Conclusion.
The equation of the circle is , which corresponds to option (C).

Final Answer:}

Step 1: Find the midpoint of the segment.
The midpoint of the line segment joining the points and is calculated using the midpoint formula:
Substituting the coordinates of the points:
Step 2: Find the slope of the line.
The slope of the line joining the points and is given by the formula:
Step 3: Find the slope of the perpendicular bisector.
The slope of the perpendicular bisector is the negative reciprocal of the slope of the line. Since the slope of the line is , the slope of the perpendicular bisector is .

Step 4: Use the point-slope form of the equation of the line.
The equation of the perpendicular bisector can be written in point-slope form as: Where is the slope and is the midpoint. Substituting , , and : Simplifying the equation: Multiply through by 2 to eliminate the fraction: Rearranging to get the equation in standard form:
Step 5: Conclusion.
The equation of the perpendicular bisector is , which corresponds to option (A).

Final Answer:}