Question

How do you write an equation for a circle given center (8,-9) and passes through (21,22)?

Answer 1

Please see the explanation for steps leading to the equation.

Explanation:

The standard form for the equation of a circle is:

`(x - h)^2 + (y - k)^2 = r^2" [1]"`

where `(x,y)` is any point on the circle, `(h,k)` is the center point, and r is the radius.

We are given that the center point is `(8, -9)`, therefore, we substitute 8 for h and -9 for k into equation [1]:

`(x - 8)^2 + (y - -9)^2 = r^2" [2]"`

NOTE: One can write `(y - -9)^2` as `(y + 9)^2` but doing so can cause errors when attempting to obtain the center point. Therefore, I do not recommend it.

To find the value of r, substitute the given point, 21 for x and 22 for y, into equation [2]:

`(21 - 8)^2 + (22 - -9)^2 = r^2`

`(13)^2 + (31)^2 = r^2`

`169 + 961 = r^2`

`r^2 = 1130`

`r = sqrt(1130)`

Substitute `sqrt(1130)` for r in equation [2]:

`(x - 8)^2 + (y - -9)^2 = (sqrt(1130))^2" [3]"`

Equation [3] is the standard form for the circle.