Question

A circle has a center that falls on the line `y = 5/6x +8` and passes through `(9 ,8 )` and `(2 ,5 )`. What is the equation of the circle?

Answer 1

The equation of the circle is `(x-68/18)^2+(y-626/57)^2=38.95`

Explanation:

Let `(a,b)` be the center of the circle and `r` the radius:
Then `b=(5a)/6+8`
The equation of the cicle is `(x-a)^2+(y-b)^2=r^2`
Now we plug in the 2 points
`(9-a)^2+(8-b)^2=r^2`
and `(2-a)^2+(5-b)^2=r^2`
`:. (9-a)^2+(8-b)^2=(2-a)^2+(5-b)^2`
`81-18a+a^2+64-16b+b^2=4-4a+a^2+25-10b+b^2`
`145-18a-16b=29-4a-10b`
`116=14a+6b` `=>` `7a+3b=58`
Solving for a and b, we get `(68/18,626/57)` as the center of the circle
Then we calculate the radius
`(2-68/18)^2+(5-626/57)^2=r^2`
`3.16+35.78=38.95=r^2`
so, the equation of tthe circle is
`(x-68/18)^2+(y-626/57)^2=38.95`