Question

In the xy -plane, the points (2,4) and (-2,-4) are the endpoints of a diameter of a circle, Which of the following is an equation of the circle?

A) (x-2)²+(y+4)²=80
B) (x-2)²+(y+4)²=20
C) x²+y²=80
D) x²+y²=20

Answer 1

The equation of the circle is x² + y² = 40.

Step-by-step explanation:

The equation of a circle in the standard form is (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r represents the radius. To find the equation of the circle given the endpoints of a diameter, we need to find the center and radius of the circle. The center of the circle is the midpoint of the diameter, which can be found by taking the average of the x-coordinates and the average of the y-coordinates of the endpoints. So, the center is ((2 + (-2))/2, (4 + (-4))/2), which simplifies to (0, 0). The radius is half the length of the diameter, which is the distance between the endpoints. Using the distance formula, we can calculate the radius as √(((-2) - 2)² + ((-4) - 4)²), which simplifies to √40 or 2√10. Substituting these values into the standard form equation, we get (x - 0)² + (y - 0)² = (2√10)², which simplifies to x² + y² = 40.