A circle has a center that falls on the line $y = 1/3x +7$ and passes through $( 3 ,1 )$ and $(6 ,9 )$ . What is the equation of the circle?

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Answer 1 · 2016-06-23T03:11:30

$color(red)((x--15/34)^2+(y-233/34)^2=26645/578)$

For the first point $(3, 1)$ we can set up one equation

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

$(3-h)^2+(1-k)^2=r^2$

For the second point $(6, 9)$ we can set up another equation

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

$(6-h)^2+(9-k)^2=r^2$
~~~~~~~~~~~~~~~~~~~~~~~~~~
$r^2=r^2$
$(3-h)^2+(1-k)^2=(6-h)^2+(9-k)^2$

After simplification, we now have an equation

$6h+16k=107$

Also from the line $y=x/3+7$ which contains the point $(h, k)$ , we have

$k=h/3+7$ or $h-3k=-21$

Simultaneous solution of the system
$h-3k=-21$
$6h+16k=107$

results to a center $(h, k)=(-15/34, 233/34)$

the radius can be computed now using center $(h, k)=(-15/34, 233/34)$ and $(6-h)^2+(9-k)^2=r^2$

$(6-h)^2+(9-k)^2=r^2$
$r^2=(6--15/34)^2+(9-233/34)^2$
$r^2=26645/578$

We can now form the equation of the circle

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

$color(red)((x--15/34)^2+(y-233/34)^2=26645/578)$

Kindly see the graph for visual inspection
![Desmos.com]( useruploads.socratic.org )

God bless....I hope the explanation is useful.

A circle has a center that falls on the line y = 1/3x +7 y = 1/3x +7 and passes through ( 3 ,1 ) ( 3 ,1 ) and (6 ,9 )(6 ,9 ). What is the equation of the circle?