Determine the center and the diameter of the circle with the equation. x^2-6x+y^2+8y=24

The center of the circle is (3, -4)

The diameter of the circle is 2

Step-by-step explanation:

The given equation of the circle is x² - 6·x + y² + 8·y = 24

The general form of the equation of a circle is x² + y² + 2·g·x + 2·f·y + c = 0

The center of the circle = (-g, -f)

The radius of the circle = √(g² + f² - c)

By comparing the given equation of the circle with the general form of the equation of a circle, we have;

2·g = -6, 2·f = 8, and c = 24

From 2·g = -6, we get;

∴ g = -6/2 = -3

g = -3

from 2·f = 8, we get;

∴ f = 8/2 = 4

f = 4

Therefore;

The center of the circle, (-g, -f) = (-(-3), -4) = (3, -4)

The center of the circle = (3, -4)

The radius of the circle, r = √(g² + f² - c) = √((-3)² + 4² - 24) = 1

The diameter of the circle, d = 2×r = 2 × 1 = 2

The diameter of the circle = 2.

step-by-step explanation: