Question

How do you find the center and radius of the circle `X^2-8x+16+y^2+10y+25=81`?

Answer 1

The given equation can be converted to the form `(x - h)^2 + (y - k)^2 = r^2` by observing the perfect squares within the equation. `(h,k)` is the center and r is the radius.

Explanation:

Given: `x^2 - 8x + 16 + y^2 + 10y + 25 = 81" [1]"`

Please observe that: `x^2 - 8x + 16 = (x - 4)^2` this makes equation [1] become:

`(x - 4)^2 + y^2 + 10y + 25 = 81" [2]"`

The y terms are, also, a perfect square: `y^2 + 10y + 25 = (y - -5)^2`

`(x - 4)^2 + (y - -5)^2 = 81" [3]"`

Finally, the constant term, `81 = 9^2`:

`(x - 4)^2 + (y - -5)^2 = 9^2`

The center is the point `(4,-5)` and the radius is 9.