A circle has a center that falls on the line #y = 2/9x +8 # and passes through # ( 3 ,1 )# and #(5 ,4 )#. What is the equation of the circle?

1 Answer
Narad T.
Nov 5, 2016

The equation of the circle is #(x+51/16)^2+(y-175/24)^2=77.87#

Explanation:

Let #(a,b)# be the center of the circle
So, #b=(2a)/9+8#
The equation of the circle is #(x-a)^2+(y-b)^2=r^2#
where #r# is the radius.
So, #(3-a)^2+(1-b)^2=r^2#
and #(5-a)^2+(4-b)^2=r^2#
#:.(3-a)^2+(1-b)^2=(5-a)^2+(4-b)^2#
#9-6a+a^2+1-2b+b^2=25-10a+a^2+16-8b+b^2#
#10-6a-2b=41-10a-8b#
#4a+6b=31#
Solving for #a and b# in the simutaneous equations, we get
#a=-51/16# and #b=175/24#
So the radius is #r^2=(3+51/16)^2+(-151/24)^2#
#r^2=(99/16)^2+(151/24)^2=77.87#
#r=8.82#
The equation of the circle is #(x+51/16)^2+(y-175/24)^2=77.87#

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