Properties of the circle (x + 9)^2 + (y - 6)^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 $$$\left[k - r, k + r\right] = \left[6 - \sqrt{102}, 6 + \sqrt{102}\right]$$$.

Answer

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

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

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