Question

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?

Answer 1

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`