A circle is given by the equation x^2+10x+y^2-8y=40. Determine the center and length of the radius

Question

New to these types of problems.

Answer 1

Complete the square for both x and y:

x² + 10x + 25 + y² - 8y + 16 = 40 + 25 + 16

Simplifying,

(x + 5)² + (y - 4)² = 81

This is now in the standard form of the equation of a circle: (x - a)² + (y - b)² = r²,

where the point (a, b) is the center and r is the radius.

Therefore, for this problem, the center is at (-5, 4) and the radius is 9.

Answer 2

x^2+10x+y^2-8y=40

(x-h)^2+(y-k)^2=r^2 is the equation of a circle

complete the square twice

10/2=5,5^2=25

-8/2=-4, (-4)^2=16

x^2+10x+25+y^2-8y+16=40+25+16

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

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

center=(-5,4)

radius=9