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Properties of the circle $$$x^{2} - 10 x + y^{2} + 6 y + 18 = 0$$$
Related calculators: Parabola Calculator, Ellipse Calculator, Hyperbola Calculator, Conic Section Calculator
Your Input
Find the center, radius, diameter, circumference, area, eccentricity, linear eccentricity, x-intercepts, y-intercepts, domain, and range of the circle $$$x^{2} - 10 x + y^{2} + 6 y + 18 = 0$$$.
Solution
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 $$$\left(x - 5\right)^{2} + \left(y - \left(-3\right)\right)^{2} = 4^{2}$$$.
Thus, $$$h = 5$$$, $$$k = -3$$$, $$$r = 4$$$.
The standard form is $$$\left(x - 5\right)^{2} + \left(y + 3\right)^{2} = 16$$$.
The general form can be found by moving everything to the left side and expanding (if needed): $$$x^{2} - 10 x + y^{2} + 6 y + 18 = 0$$$.
Center: $$$\left(5, -3\right)$$$.
Radius: $$$r = 4$$$.
Diameter: $$$d = 2 r = 8$$$.
Circumference: $$$C = 2 \pi r = 8 \pi$$$.
Area: $$$A = \pi r^{2} = 16 \pi$$$.
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).
x-intercepts: $$$\left(5 - \sqrt{7}, 0\right)$$$, $$$\left(\sqrt{7} + 5, 0\right)$$$
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 $$$\left[h - r, h + r\right] = \left[1, 9\right]$$$.
The range is $$$\left[k - r, k + r\right] = \left[-7, 1\right]$$$.
Answer
Standard form/equation: $$$\left(x - 5\right)^{2} + \left(y + 3\right)^{2} = 16$$$A.
General form/equation: $$$x^{2} - 10 x + y^{2} + 6 y + 18 = 0$$$A.
Graph: see the graphing calculator.
Center: $$$\left(5, -3\right)$$$A.
Radius: $$$4$$$A.
Diameter: $$$8$$$A.
Circumference: $$$8 \pi\approx 25.132741228718346$$$A.
Area: $$$16 \pi\approx 50.265482457436692$$$A.
Eccentricity: $$$0$$$A.
Linear eccentricity: $$$0$$$A.
x-intercepts: $$$\left(5 - \sqrt{7}, 0\right)\approx \left(2.354248688935409, 0\right)$$$, $$$\left(\sqrt{7} + 5, 0\right)\approx \left(7.645751311064591, 0\right)$$$A.
y-intercepts: no y-intercepts.
Domain: $$$\left[1, 9\right]$$$A.
Range: $$$\left[-7, 1\right]$$$A.