How do you write an equation for a circle given endpoints of a diameter at (-5,2) and (3,6)?

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Answer 1 · 2016-10-24T04:58:50

#(x+1)^2+(y-4)^2=20#

First, we have to find the centre of the circle.

This will be halfway between the two endpoints

#(3+(-5))/2 =-1#, #(6+2)/2=4#

giving us the point #(-1,4)# for the centre

now we need to find the radius,
take the middle and an end point and find the distance,

#((3)-(-1)# , #(6)-(4))#

giving us #(4,2)# or 4 our units horizontal and two vertical.

now if we use trigonometry to find the radius

#r=sqrt(4^2+2^2) = sqrt(20)#

now the equation of a circle is,

#(x-a)^2+(y-b)^2=r^2#

where a and b are the coordinates of the centre.
sub in,

#(x-(-1))^2+(y-(4))^2=(sqrt(20))^2#

#(x+1)^2+(y-4)^2=20#

giving us,

#(x+1)^2+(y-4)^2=20#

for the equation of our circle.