Question

How do you write an equation of a circle with center (2, 4) and passes through the point (6,7)?

Answer 1

`(x-2)^2+(y-4)^2=25`

Explanation:

If the circle has its center at `(x_c,y_c)=(2,4)` and passes through `(6,7)`
then its radius is
`color(white)("XXX")r=sqrt((6-2)^2+(7-3)^2) =5`

Since the general equation for a circle with center `(x_c,y_c)` and radius `r` is
`color(white)("XXX")(x-x_c)^2+(y-y_c)^2=r^2`

the required circle has an equation:
`color(white)("XXX")(x-2)^2+(y-4)^2=5^2`

I find this version the most informative, but your instructor may prefer, the standard polynomial form:
`color(white)("XXX")x^2+y^2-4x-8y-5=0`
which can ber derived by expanding and simplifying the standard circle equation form.