Question

What is the standard form of the equation of a circle with center (1,2) intersects x-axis at -1 and 3?

Answer 1

`(x-1)^2+(y-2)^2=8`

Explanation:

The general standard form of the equation for a circle with center `(a,b)` and radius `r` is
`color(white)("XXX")(x-a)^2+(y-b)^2=r^2`

In the case the radius is the distance between the center `(1,2)` and one of the points on the circle; in this case we could use either of the x-intercepts: `(-1,0)` or `(3,0)`
to get (using `(-1,0)`):
`color(white)("XXXXXXXX")r=sqrt((1-(-1))^2+(2-0)^2)=2sqrt(2)`

Using `(a,b) = (1,2)` and `r^2=(2sqrt(2))^2 = 8`
with the general standard form gives the answer above.