Hello, caramelcake!

Find the equation of the circle whose center is on $\displaystyle y \:=\:6-2x$

and which passes through $\displaystyle A(\text{-}2,0)$ and $\displaystyle B(4,0).$

Since the $\displaystyle y$-value of both points $\displaystyle A$ and $\displaystyle B$ are 0,

I can gather that the diameter is 6 units. . This is not true!

Did you make a sketch?

Code:

\ |
\ * * *
*\ | *
* \ | : *
* \| : *
* :
* |\:C *
* | o *
* | :\ *
| : \
* | : \ *
- - o - - + - - \ - o - - - - -
-2 * | : \* 4
* * * \
| :

The $\displaystyle x$-intercepts of the circle are: $\displaystyle A(\text{-}2,0)$ and $\displaystyle B(4,0)$

The center lies on the vertical line midway between the intercepts: .$\displaystyle x \:=\:1$

The center also lies on the line $\displaystyle y \:=\:6-2x$

. . The center is the intersection of the two lines: .$\displaystyle C(1,4)$

The radius is the distance $\displaystyle \overline{CA}\!:\;5 $

Therefore: .$\displaystyle (x-1)^2 + (y-4)^2 \:=\:5^2$