Question

The equation of the circle whose end points of a diameter are the centres of the circles \[ x^2 + y^2 + 2x - 4y + 1 = 0 \quad \text{and} \quad x^2 + y^2 - 8x + 6y + 17 = 0 \] is

• A. \( x^2 + y^2 - 3x - y - 10 = 0 \)
• B. \( x^2 + y^2 + 3x - y - 10 = 0 \)
• C. \( x^2 + y^2 + 3x - y - 10 = 0 \)
• D. \( x^2 + y^2 - 3x + y - 10 = 0 \)

Hint:

To find the equation of a circle with given diameter endpoints, first find the midpoint (center) and use the distance between the endpoints to find the radius.

Answer 1

The Correct Option is D

Step 1: Find the midpoints of the diameter.
The center of a circle is the midpoint of the ends of its diameter. The midpoint of the given points of the diameters gives us the center of the required circle. Using the midpoint formula for the centers of the given circles, we obtain the center coordinates.
Step 2: Use the center and radius to write the equation of the required circle.
We now use the general form of the circle's equation and plug in the calculated center and radius. We arrive at the equation: \[ x^2 + y^2 - 3x + y - 10 = 0 \]
Step 3: Conclusion.
Thus, the equation of the required circle is \( x^2 + y^2 - 3x + y - 10 = 0 \), corresponding to option (D).