A circle has a center that falls on the line #y = 8/9x +7 # and passes through # ( 4 ,1 )# and #(3 ,7 )#. What is the equation of the circle?

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Answer 1 · 2018-05-15T17:21:08

#color(blue)((x+129/26)^2+(y-101/39)^2=503977/6084)#

The general equation of a circles is:

#(x-h)^2+(y-k)^2=r^2#

Where #(h,k)# are the #x and y# coordinates of the centre respectively, and r is the radius.

So #(h,k)# are on the line #y=8/9x+7#

Substituting #h# and #k#:

#k=8/9h+7 \ \ \ \ \ \[1]#

#(4,1)# lie on the circumference:

#(4-h)^2+(1-k)^2=r^2 \ \ \ \ \[2]#

#(3,7)# lie on the circumference:

#(3-h)^2+(7-k)^2=r^2 \ \ \ \[3]#

Subtracting #[3]# from #[2]#

#-41-2h+12k=0 \ \ \ \[4]#

Substituting #[1]# into #[4]#

#-41-2h+12(8/9h+7)=0#

#43+26/3h=0#

#h=-129/26#

Substituting in #[1]#

#k=8/9(-129/26)+7=101/39#

Substituting #h# and #k# in #[2]#

#(4+129/26)^2+(1-101/39)^2=r^2=503977/6084#

Equation is:

#(x+129/26)^2+(y-101/39)^2=503977/6084#

GRAPH:

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