Properties of the circle $$$x^{2} + 8 x + y^{2} - 6 y + 21 = 0$$$

The standard form of the equation of a circle is $$$\left(x - h\right)^{2} + \left(y - k\right)^{2} = r^{2}$$$, where $$$\left(h, k\right)$$$ is the center of the circle and $$$r$$$ is the radius.

Our circle in this form is $$$\left(x - \left(-4\right)\right)^{2} + \left(y - 3\right)^{2} = 2^{2}$$$.

Thus, $$$h = -4$$$, $$$k = 3$$$, $$$r = 2$$$.

The standard form is $$$\left(x + 4\right)^{2} + \left(y - 3\right)^{2} = 4$$$.

The general form can be found by moving everything to the left side and expanding (if needed): $$$x^{2} + 8 x + y^{2} - 6 y + 21 = 0$$$.

Center: $$$\left(-4, 3\right)$$$.

Radius: $$$r = 2$$$.

Diameter: $$$d = 2 r = 4$$$.

Circumference: $$$C = 2 \pi r = 4 \pi$$$.

Area: $$$A = \pi r^{2} = 4 \pi$$$.

Both eccentricity and linear eccentricity of a circle equal $$$0$$$.

The x-intercepts can be found by setting $$$y = 0$$$ in the equation and solving for $$$x$$$ (for steps, see intercepts calculator).

Since there are no real solutions, there are no x-intercepts.

The y-intercepts can be found by setting $$$x = 0$$$ in the equation and solving for $$$y$$$: (for steps, see intercepts calculator).

Since there are no real solutions, there are no y-intercepts.

The domain is $$$\left[h - r, h + r\right] = \left[-6, -2\right]$$$.

The range is $$$\left[k - r, k + r\right] = \left[1, 5\right]$$$.

Standard form/equation: $$$\left(x + 4\right)^{2} + \left(y - 3\right)^{2} = 4$$$A.

General form/equation: $$$x^{2} + 8 x + y^{2} - 6 y + 21 = 0$$$A.

Graph: see the graphing calculator.

Center: $$$\left(-4, 3\right)$$$A.

Radius: $$$2$$$A.

Diameter: $$$4$$$A.

Circumference: $$$4 \pi\approx 12.566370614359173$$$A.

Area: $$$4 \pi\approx 12.566370614359173$$$A.

Eccentricity: $$$0$$$A.

Linear eccentricity: $$$0$$$A.

x-intercepts: no x-intercepts.

y-intercepts: no y-intercepts.

Domain: $$$\left[-6, -2\right]$$$A.

Range: $$$\left[1, 5\right]$$$A.