Question

What are the coordinates of the center of the circle `x^2 + y^2 -8x -10y -8=0`?

Answer 1

`(4, 5)`

Explanation:

Complete the squares for `x` and `y` as follows:

`0 = x^2+y^2-8x-10y-8`

`= color(blue)(x^2-8x+16)+color(green)(y^2-10y+25)-49`

`= color(blue)((x-4)^2)+color(green)((y-5)^2)-7^2`

Add `7^2` to both ends and transpose to get:

`(x-4)^2+(y-5)^2 = 7^2`

This is in the form of the standard equation of a circle:

`(x-h)^2+(y-k)^2 = r^2`

with center `(h, k) = (4, 5)` and radius `r=7`.

graph{((x-4)^2+(y-5)^2-7^2)((x-4)^2+(y-5)^2-0.06)=0 [-16.67, 23.33, -5.2, 14.8]}