Write the equation of a circle with endpoints of the diameter at (2, -5) and (-4, 3).?

Question

Write the equation of a circle with endpoints of the diameter at (2, -5) and (-4, 3).
Show your work for credit.

Answer 1

Since those are the endpoints of the diameter, then their midpoint is the center.
The midpoint is ((2+-4)/2, (-5+3)/2) = (-2/2, -2/2) = (-1,-1)

The length of the diameter is half the radius.
d = sqrt [(2- -4)^2 + (-5-3)^2]
d = sqrt [(6)^2 + (-8)^2]
d = sqrt(36 + 64)
d = sqrt(100)
d = 10
so r = 10/2 = 5

The equation of a circle is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center and r is the radius.
(x+1)^2 + (y+1)^2 = 25

Answer 2

A circle is set of points equidistant from its center and the line from any point on the circle to center is radius.
let center lies at (a, b) and radius is r
Let any point on the circle be (x, y)
distance between (x, y) and (a,b) =sqrt[ (x -a)^2 + (y -b)^2]
so sqrt [ (x-a)^2 + (y -b)^2] = r
As end points of diameter are given , its mid point lies at the center
the co ordinates of center (a,b) = ((2 - 4)/2, (-5 + 3)/2)
= (-1, -1)
(diameter/2)^2 = (radius)^2
(diameter)^2/4 = (radius)^2
[(2 - (-4))^2 + (-5 - 3)^2]/4 =r^2
[6^2 + (-8)^2]/4 =r^2
(36 + 64)/4 = r^2
100/4 = r^2
r^2 = 25
substituting values of a, b and r^2 in eqn (1)
(x - (-1) )^2 + (y - (-1))^2 = 25
(x +1)^2 + (y + 1)^2 = 25

Answer 3

The center is at (-1, -1). Radius = sqrt(3^2 + 4^2) = 5

(y +1)^2 + (x + 1)^2 = 25

Answer 4

Centre of the circle = {[2 + (-4)]*0.5, [-5 + 3]*0.5} = (-1, -1)
Radius of the circle = sqrt{[(2- (-4)]^2 + [-5 -3]^2} / 2= 5

Therefore the equation of the circle

[x - (-1)]^2 + [y - (-1)]^2 = 25
x^2 + y^2 +2x + 2y -23 = 0

Answer 5

the center is (2-4/2=-1 , -5+3/2=-1) i.e. (-1,-1)
the diameter length= sq.root (2+4)^2+(-5-3)^2 = 10 then r=5
then the eqn is (x+1)^2+(y+1)^2=25
x^2+y^2 +2x +2y-23=0

Answer 6

midpoint gives you center ...(-1, -1)


(x+1)^2 + (y+1)^2 = 25

does (2,-5) to (-4,3) = 10
(6)^2 + (8)^2 = 10^2 (yes)

Answer 7

http://www.westirondequoit.org/ihs/Math/mathamastery/pdf_files/eq_circ_endpt_diameter/WriteEQCirDiameterModels.pdf