How do you find the center, radius, and equation of a circle with : (1,-2) and (7,6) as the diameter's endpoints?

1 Answer
Jan 26, 2016

Answer:

#(x-4)^2 +(y-2)^2 = 25#

Explanation:

The centre of the circle #c# is at the midpoint of the diameter, and the radius #r#is equal to half the diameter. We therefore need the distance between the two points.
Sketch
Using Pythagoras to calculate the length of the diameter #xy# gives
#(xy)^2 = (7-1)^2 + (6-(-2))^2#
#(xy)^2 = 36 +64 =100#

#:. yx =sqrt(100) = 10#

The radius is therefore #5#

#c# is at the midpoint of #xy#. Its #x# coordinate #h# is therefore midway between #x# and #z#, and its #y# coordinate #k# is midway between #z# and #y#.
#h = 1 + (7-1)/2 = 4#
#k = -2 + (6-(-2))/2 = 2#

The equation of the circle is therefore #(x-4)^2 +(y-2)^2 = 25#