Question

The end-points of a diameter of a circle are \( (-1, 4) \) and \( (5, 4) \). Then the equation of the circle is

• A. \( (x - 3)^2 + y^2 = 9 \)
• B. \( (x - 3)^2 + (y + 4)^2 = 3 \)
• C. \( (x - 2)^2 + (y - 4)^2 = 9 \)
• D. \( (x + 3)^2 + (y + 4)^2 = 9 \)
• E. \( (x - 3)^2 + (y - 4)^2 = 4 \)

Hint:

The equation of a circle can be derived from the center and radius. The center is the midpoint of the diameter, and the radius is the distance from the center to any endpoint of the diameter.

Answer 1

The Correct Option is C

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 \( (-1, 4) \) and \( (5, 4) \) is calculated using the midpoint formula:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting the coordinates of the points: \[ \text{Midpoint} = \left( \frac{-1 + 5}{2}, \frac{4 + 4}{2} \right) = \left( 2, 4 \right) \] Thus, the center of the circle is \( (2, 4) \).

Step 2: Calculate the radius of the circle.
The radius is the distance from the center of the circle \( (2, 4) \) to either endpoint of the diameter. We can use the distance formula to find this distance. Using the point \( (-1, 4) \) as one endpoint: \[ \text{Radius} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting \( (x_1, y_1) = (2, 4) \) and \( (x_2, y_2) = (-1, 4) \): \[ \text{Radius} = \sqrt{(-1 - 2)^2 + (4 - 4)^2} = \sqrt{(-3)^2 + 0^2} = \sqrt{9} = 3 \] Thus, the radius is 3.

Step 3: Write the equation of the circle.
The general equation of a circle with center \( (h, k) \) and radius \( r \) is: \[ (x - h)^2 + (y - k)^2 = r^2 \] Substituting \( h = 2 \), \( k = 4 \), and \( r = 3 \): \[ (x - 2)^2 + (y - 4)^2 = 9 \]
Step 4: Conclusion.
The equation of the circle is \( (x - 2)^2 + (y - 4)^2 = 9 \), which corresponds to option (C).
\[ \boxed{(x - 2)^2 + (y - 4)^2 = 9} \]
Final Answer:} \( (x - 2)^2 + (y - 4)^2 = 9 \)