Given A = (-1,0) and B = (11,4), how do you show that the equation of the circle with AB as diameter may be written as #(x-5)^2 + (y-2)^2 = 40#?

1 Answer
Dec 23, 2016

Answer:

see explanation.

Explanation:

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Recall that the center-radius form of the circle equation is in the format:
#(x – h)^2 + (y – k)^2 = r^2#,
with the center being at the point #(h, k)# and the radius being #r#.

Given #A(-1,0), and B(11,4)#,
Let distance between #A and B# be #D#.
#=> D=sqrt((11-(-1))^2+(4-0)^2)=sqrt160=sqrt(16*10)=4sqrt10#
Given #D# is the diameter, #=> D/2 = R# (radius)
#=> R=D/2=(4sqrt10)/2=2sqrt10#
The circle is centered at midpoint of AB :
#=># midpoint of #AB= ((-1+11)/2, (0+4)/2)=(5,2)#

So the equation of the circle can be written as :
#(x-5)^2+(y-2)^2=(2sqrt10)^2#,
#=> (x-5)^2+(y-2)^2=40# (proved)