# Properties of the circle $\left(x + 9\right)^{2} + \left(y - 6\right)^{2} = 102$

The calculator will find the properties of the circle $\left(x + 9\right)^{2} + \left(y - 6\right)^{2} = 102$, with steps shown.

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Find the center, radius, diameter, circumference, area, eccentricity, linear eccentricity, x-intercepts, y-intercepts, domain, and range of the circle $\left(x + 9\right)^{2} + \left(y - 6\right)^{2} = 102$.

### 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 $\left(x - \left(-9\right)\right)^{2} + \left(y - 6\right)^{2} = \left(\sqrt{102}\right)^{2}$.

Thus, $h = -9$, $k = 6$, $r = \sqrt{102}$.

The standard form is $\left(x + 9\right)^{2} + \left(y - 6\right)^{2} = 102$.

The general form can be found by moving everything to the left side and expanding (if needed): $x^{2} + 18 x + y^{2} - 12 y + 15 = 0$.

Center: $\left(-9, 6\right)$.

Radius: $r = \sqrt{102}$.

Diameter: $d = 2 r = 2 \sqrt{102}$.

Circumference: $C = 2 \pi r = 2 \sqrt{102} \pi$.

Area: $A = \pi r^{2} = 102 \pi$.

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: $\left(-9 - \sqrt{66}, 0\right)$, $\left(-9 + \sqrt{66}, 0\right)$

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

y-intercepts: $\left(0, 6 - \sqrt{21}\right)$, $\left(0, \sqrt{21} + 6\right)$

The domain is $\left[h - r, h + r\right] = \left[- \sqrt{102} - 9, -9 + \sqrt{102}\right]$.

The range is $\left[k - r, k + r\right] = \left[6 - \sqrt{102}, 6 + \sqrt{102}\right]$.

Standard form/equation: $\left(x + 9\right)^{2} + \left(y - 6\right)^{2} = 102$A.

General form/equation: $x^{2} + 18 x + y^{2} - 12 y + 15 = 0$A.

Graph: see the graphing calculator.

Center: $\left(-9, 6\right)$A.

Radius: $\sqrt{102}\approx 10.099504938362078$A.

Diameter: $2 \sqrt{102}\approx 20.199009876724156$A.

Circumference: $2 \sqrt{102} \pi\approx 63.457061038504283$A.

Area: $102 \pi\approx 320.44245066615891$A.

Eccentricity: $0$A.

Linear eccentricity: $0$A.

x-intercepts: $\left(-9 - \sqrt{66}, 0\right)\approx \left(-17.12403840463596, 0\right)$, $\left(-9 + \sqrt{66}, 0\right)\approx \left(-0.87596159536404, 0\right)$A.

y-intercepts: $\left(0, 6 - \sqrt{21}\right)\approx \left(0, 1.41742430504416\right)$, $\left(0, \sqrt{21} + 6\right)\approx \left(0, 10.58257569495584\right)$A.

Domain: $\left[- \sqrt{102} - 9, -9 + \sqrt{102}\right]\approx \left[-19.099504938362078, 1.099504938362078\right].$A

Range: $\left[6 - \sqrt{102}, 6 + \sqrt{102}\right]\approx \left[-4.099504938362078, 16.099504938362078\right].$A