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:

Diametric form: $(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0$.
$(x+1)(x-5) + (y-4)(y-4) = 0 \Rightarrow x^2 - 4x - 5 + (y-4)^2 = 0 \Rightarrow (x-2)^2 - 4 - 5 + (y-4)^2 = 0$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The equation of a circle in standard center-radius form is \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.\ The center of the circle is the midpoint of its diameter. Step 2: Key Formula or Approach:
1. Center \((h, k) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).\ 2. Radius \(r\) is half the diameter length or the distance from the center to one of the endpoints. Step 3: Detailed Explanation:
Let the given diameter endpoints be \(A(-1, 4)\) and \(B(5, 4)\).\ First, calculate the center \((h, k)\): \[ h = \frac{-1 + 5}{2} = \frac{4}{2} = 2 \] \[ k = \frac{4 + 4}{2} = \frac{8}{2} = 4 \] So, the center is \((2, 4)\). Next, calculate the radius \(r\) as the distance from center \((2, 4)\) to endpoint \(B(5, 4)\): Since the y-coordinates are the same, the distance is horizontal: \[ r = \sqrt{(5 - 2)^2 + (4 - 4)^2} = \sqrt{3^2} = 3 \] Now substitute the center and radius into the circle equation: \[ (x - h)^2 + (y - k)^2 = r^2 \] \[ (x - 2)^2 + (y - 4)^2 = 3^2 \] \[ (x - 2)^2 + (y - 4)^2 = 9 \] Step 4: Final Answer:
The correct equation is \( (x - 2)^2 + (y - 4)^2 = 9 \).