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 + y² = 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

Answer 1

The Correct Option is C

Step 1: Find the center of the circle  
The center of a circle whose diameter has endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Given endpoints: \((-1,4)\) and \( (5,4) \) \[ \text{Center} = \left( \frac{-1 + 5}{2}, \frac{4 + 4}{2} \right) = \left( \frac{4}{2}, \frac{8}{2} \right) = (2,4) \]

Step 2: Find the radius 
The radius is half the length of the diameter. The length of the diameter is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] \[ = \sqrt{(5 - (-1))^2 + (4 - 4)^2} = \sqrt{(5 + 1)^2 + 0} = \sqrt{6^2} = 6 \] Radius \( r \) is: \[ r = \frac{6}{2} = 3 \]

Step 3: Write the equation of the circle 
The standard equation of a circle is: \[ (x - h)^2 + (y - k)^2 = r^2 \] Substituting \( (h, k) = (2,4) \) and \( r = 3 \): \[ (x - 2)^2 + (y - 4)^2 = 9 \]

Final Answer: The equation of the circle is \((x - 2)^2 + (y - 4)^2 = 9\).