The endpoints of a diameter of a circle are (-3,2) and (5,-6). find the center and radius of the circle. write the equation of the circle.

Given the endpoints of the circle

(-3,2) (5,-6)

x1 = -3

x2 = 5

y1 = 2

y2 = -6

Determine the center, the radius and the equation of the circle.

Solution:

A. Center of the Circle
To determine the center of the circle we used the midpoint formula.

Midpoint formula =(x1 + x2)/2 , (y1 + y2) /2

Substitute the given values of x1, x2 , y1 and y2 on the formula for the midpoint.

Midpoint = (-3 + 5) /2 , ( 2 - 6)/ 2

Midpoint = 2/2 , -4 /2

Midpoint= 1 , -2

The center of the circle is located at ( 1, -2)

B. Radius of the Circle

Next we determine the radius of the circle, in order to do so, we subtract y1 and y2 by the value of y from the center of the circle.

r = 2 -(-2)

r = 4

r = |-6 -2|

r= |-4|

r= 4

we've used absolute value since we cannot have a negative radius.

The radius of the circle is 4.

C. Equation of the Circle

In order to determine the equation of the circle we look at formula structure of the equations.

Formula of the circle when the center is at the origin.

x^2 + y^2 = r^2

Formula of the circle when the center is at any point (h,k)

(x-h)^2 + (y-k)^2 = r^2

Since we are given the points (1, -2) then we will use the second formula.

(x-1)^2 + (y + 2) ^2 = 4^2

Expand and combine like terms.

x^2 -2x + 1 + y^2 + 4y + 4 = 16

x^2 + y^2  - 2x + 4y + 5 - 16 = 0

x^2 + y^2  - 2x + 4y -11 = 0

The equation of the circle is x^2 + y^2  - 2x + 4y -11 = 0