Points E(-2, -1) and F(3, 2) are the ends of the diameter of a circle. Find the equation of the circle.
-
A.
(x^{2} + y^{2} – 5x + 3 = 0) -
B.
(x^{2} + y^{2} – 2x – 6y – 13 = 0) -
C.
(x^{2} + y^{2} – x + 5y – 6 = 0) -
D.
(x^{2} + y^{2} – x – y – 8 = 0)
Correct Answer: Option D
Explanation
Given the endpoints of the diameter |EF|, the midpoint is the centre of the circle
= ((frac{-2 + 3}{2} , frac{-1 + 2}{2}) = (frac{1}{2} , frac{1}{2}))
The radius is the distance from the centre to any point on the circle. Using ((frac{1}{2}, frac{1}{2})) and ((3, 2));
(r^{2} = (3 – frac{1}{2})^{2} + (2 – frac{1}{2})^{2} = frac{25}{4} + frac{9}{4})
(r^{2} = frac{34}{4})
The equation of a circle is given as:
((x – a)^{2} + (y – b)^{2} = r^{2}), (a, b) as the centre of the circle.
(= (x – frac{1}{2})^{2} + (y – frac{1}{2})^{2} = frac{34}{4})
(x^{2} – x + frac{1}{4} + y^{2} – y + frac{1}{4} = frac{17}{2})
= (x^{2} – y^{2} – x – y – 8 = 0)