Answer:
The center is ( -5, 8) and radius is 13.
Step-by-step 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.
Answer:
Center: (-5,8)
Radius: 13
Step-by-step explanation:
The equation of the circle in center-radius form is:
[tex](x-h)^2+(y-k)^2=r^2[/tex]
Where the point (h,k) is the center of the circle and "r" is the radius.
Subtract 16y from both sides of the equation:
[tex]-10x+80+16y-16y=x^2+y^2-16y\\\\-10x+80=x^2+y^2-16y[/tex]
Add 10x to both sides:
[tex]-10x+80+10x=x^2+y^2-16y+10x\\\\80=x^2+y^2-16y+10x[/tex]
Make two groups for variable "x" and variable "y":
[tex](x^2+10x)+(y^2-16y)=-80[/tex]
Complete the square:
Add [tex](\frac{10}{2})^2=5^2[/tex] inside the parentheses of "x".
Add [tex](\frac{16}{2})^2=8^2[/tex] inside the parentheses of "y".
Add [tex]5^2[/tex] and [tex]8^2[/tex] to the right side of the equation.
Then:
[tex](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[/tex]
We know that [tex]\sqrt{169}=13[/tex]
Then, rewriting, you get that the equation of the circle in center-radius form is:
[tex](x+5)^2+(y-8)^2=13^2[/tex]
You can observe that the radius of the circle is:
[tex]r=13[/tex]
And the center is:
[tex](h,k)=(-5,8)[/tex]
