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

To find the equation of a circle when given the endpoints of its diameter, we use the following steps:

1. Identify the midpoint of the diameter as the center of the circle.
2. Calculate the radius using the distance formula between the given endpoints.
3. Use the standard form of the equation of a circle, \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center, and \(r\) is the radius.

Let's perform these steps for the endpoints \((−1, 4)\) and \((5, 4)\):

1. Find the center of the circle:
   • The midpoint of the diameter is the average of the coordinates of the endpoints.
   • Midpoint, \((h, k) = \left(\frac{-1 + 5}{2}, \frac{4 + 4}{2}\right) = (2, 4)\).
2. Calculate the radius:
   • The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
   • Here, distance (diameter) = \(\sqrt{(5 - (-1))^2 + (4 - 4)^2} = \sqrt{(6)^2} = 6\).
   • Radius, \(r = \frac{6}{2} = 3\).
3. Write the equation of the circle:
   • The circle's equation is \((x - 2)^2 + (y - 4)^2 = 3^2\).
   • Simplified equation: \((x - 2)^2 + (y - 4)^2 = 9\).

Conclusion: Therefore, the equation of the circle is \((x − 2)^2 + (y − 4)^2 = 9\), which matches the correct option.