Question

Identify the center and radius -10x+80+16y=x^2+y^2

Answer 1

Center: (-5,8)

Radius: 13

Explanation:

The equation of the circle in center-radius form is:

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

Where the point (h,k) is the center of the circle and "r" is the radius.

Subtract 16y from both sides of the equation:

`-10x+80+16y-16y=x^2+y^2-16y\\\\-10x+80=x^2+y^2-16y`

Add 10x to both sides:

`-10x+80+10x=x^2+y^2-16y+10x\\\\80=x^2+y^2-16y+10x`

Make two groups for variable "x" and variable "y":

`(x^2+10x)+(y^2-16y)=-80`

Complete the square:

Add
`((10)/(2))^2=5^2` inside the parentheses of "x".

Add
`((16)/(2))^2=8^2` inside the parentheses of "y".

Add
`5^2` and
`8^2` to the right side of the equation.

Then:

`(x^2+10x+5^2)+(y^2-16y+8^2)=80+5^2+8^2\\\\(x^2+10x+5^2)+(y^2-16y+8^2)=169`

We know that
`√(169)=13`

Then, rewriting, you get that the equation of the circle in center-radius form is:

`(x+5)^2+(y-8)^2=13^2`

You can observe that the radius of the circle is:

`r=13`

And the center is:

`(h,k)=(-5,8)`

Answer 2

The center is ( -5, 8) and radius is 13.

Explanation:

The center of a circle is given (h,k) and the radius is r. The formula is

(x-h)²+ (y-k)²=r², so we need to express our given into that form.

To start, -10x+80+16y=x^2+y^2

80 = x²+ 10x + y²-16y or x²+ 10x + y²-16y = 80

we need to use the steps in completing the square

x²+ 10x + _____ + y²-16y + _____ = 80+ ____+_____

use (b/2)² in the blanks

x²+ 10x + 25 + y²-16y + 64 = 80+ 25+64

on the left side of the equation factor them while simplify the right side

(x- (-5))²+ (y-8)² = 169.

Now our equation is in the form of (x-h)²+ (y-k)²=r²

so h = -5 k = 8 and r is 13.