Question

A circle has a center that falls on the line `y = 7/6x +1` and passes through `(9 ,8 )` and `(2 ,5 )`. What is the equation of the circle?

Answer 1

`(x - 110/21)^2 + (y - 896/126)^2 = 14.94`

Explanation:

We have a point (x, y) that is the center of the circle.
This means that the distance from points (9,8) and (2,5) to this point are the same.
Thanks to our pal Mr. Pythagoras, we know that:
From (9,8) to (x,y) the distance is
`sqrt((9-x)^2 + (8-y)^2)`
and from (2,5) the distance is:
`sqrt((2-x)^2 + (5-y)^2)`

and since these evaluate to the same radius, we can write this equation:

`(9 - x)^2 + (8-y)^2 = (2-x)^2 + (5-y)^2`

multiply everything out:

`81 - 18x + x^2 + 64 -16y + y^2 = 4-4x + x^2 + 25 - 10y + y^2`

We can subtract `x^2` and `y^2` from each side, and collect the terms:
`145 -18x -16y = 29 - 4x -10y`

...subtract the terms on the right from the terms on the left, and we have:
`116 -14x - 6y = 0`

...and now we use the other piece of our information from the problem statement: `y = 7/6x + 1`. We substitue this value of y into the equation above:

`116 - 14x - 6(7/6x + 1) = 0`
`116 - 14x - 7x -6 = 0`
`110 - 21x = 0`
`110 = 21x`
`x = 110/21`

...we've solved for x, so we can use the equation for the line we were given to solve for y:

`y = 7/6(110/21) + 1 = 770/126 + 1 = 770/126 + 126/126 = 896/126`

So we have the coordinates of the center point. We can use these to construct the equation of the circle.

We need to know the radius. This is just the distance from the center to the points we were given. We can use either of these to calculate the radius:
`r = sqrt((9 - 110/21)^2 + (8 - 896/126)^2)`

...I make this about 3.865
(I encourage you to calculate the distance to the other point (2,5) as a check. The distance must be the same or very close, differing only by rounding error).

We actually need `r^2` for our circle equation - this is 14.94 (rounding).

So we can write our circle equation:
`(x - 110/21)^2 + (y - 896/126)^2 = 14.94`

GOOD LUCK