If (2,-7) and (8,15) are the endpoints of a diameter of a circle, what is the equation of the circle

Given the endpoints of a diameter of a circle:

(2, -7) (8, 15)

x1 = 2

x2 = 8

y1= -7

y2 = 15

Determine the equation of the circle.

The equation of a circle can be presented as,

Center at the origin: x^2 + y^2 = r^2

Center at any points (h,k): (x-h)^2 + (y - k)^2 =r^2

Solution:

In order to determine the equation of the circle, since we are given the diameter, we must first find the midpoint.

midpoint formula (x1 + x2)/2 , (y1 + y2)/2

Midpoint: (2 + 8)/2 , (-7 + 15) /2

Midpoint: 10/2 , 8/2

Midpoint: (5 , 4)

Now we need to determine the radius of the circle, this can be done by subtracting the midpoint y, to the endpoints y1, and y2

r = 15 - 4

r = 11

r = | -7 -4|

r = |-11|

We've used absolute value since we are finding the exact number of the radius and it cannot be negative. Hence, absolute value of any negative number is a positive number

r = 11

Now we can write the equation of the circle since we have the midpoint and the radius. We will use the equation number 2 as the basis, since the center is not at the origin.

Midpoint (5,4) 

r=11

(x-5)^2 + (y-4)^2 = 11^2

(x-5)^2 + (y-4)^2 = 121

Final answer:

(x-5)^2 + (y-4)^2 = 121