Question

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

Answer 1

`(x-19)^2+(y-40/3)^2=2665/9`

Explanation:

From the given two points `(3, 7)` and `(7, 1)` we will be able to establish equations

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

`(3-h)^2+(7-k)^2=r^2" "`first equation using `(3, 7)`

and

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

`(7-h)^2+(1-k)^2=r^2" "`second equation using `(7, 1)`

But `r^2=r^2`
therefore we can equate first and second equations

`(3-h)^2+(7-k)^2=(7-h)^2+(1-k)^2`

and this will be simplified to
`h-3k=-2" "`third equation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The center `(h, k)` passes thru the line `y=1/3x+7` so we can have an equation

`k=1/3h+7` because the center is one of its points

Using this equation and the third equation,

`h-3k=-2" "`
`k=1/3h+7`

The center `(h, k)=(19, 40/3)` by simultaneous solution.

We can use the equation
`(3-h)^2+(7-k)^2=r^2" "`first equation
to solve for the radius `r`

`r^2=2665/9`

and the equation of the circle is

`(x-19)^2+(y-40/3)^2=2665/9`

Kindly see the graph to verify the equation of the circle `(x-19)^2+(y-40/3)^2=2665/9` colored red, with points `(3, 7)` colored green, and `(7, 1)` colored blue, and the line `y=1/3x+7` colored orange which contains the center `(19, 40/3)` colored black.

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