Properties of the hyperbola -(x - 6)^2 + (y - 5)^2/64 = 1

The calculator will find the properties of the hyperbola $$$- \left(x - 6\right)^{2} + \frac{\left(y - 5\right)^{2}}{64} = 1$$$, with steps shown.

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Equation:

Center:$$$($$$

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First focus:$$$($$$

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Second focus:$$$($$$

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First vertex:$$$($$$

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Second vertex:$$$($$$

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First co-vertex:$$$($$$

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Second co-vertex:$$$($$$

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Length of the major (transverse) axis:

If you have the length of the semi-major axis (a), enter its value multiplied by $$$2$$$ here.

Length of the minor (conjugate) axis:

If you have the length of the semi-minor axis (b), enter its value multiplied by $$$2$$$ here.

Eccentricity:

First directrix:

In any form you want: $$$y = -6$$$, $$$y = 2 x + 5$$$, etc.

Second directrix:

In any form you want: $$$x - 3 = 0$$$, $$$- 8 x + 7 y + 1 = 0$$$, etc.

First asymptote:

In any form you want: $$$x = -2$$$, $$$y = 5 - 3 x$$$, etc.

Second asymptote:

In any form you want: $$$y + 1 = 0$$$, $$$x = 2 y - 7$$$, etc.

First point:$$$($$$

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Second point:$$$($$$

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Third point:$$$($$$

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Fourth point:$$$($$$

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Vertical major (transverse) axis (parallel to the y-axis)?

Horizontal minor (conjugate) axis (parallel to the x-axis)?

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Find the center, foci, vertices, co-vertices, major axis length, semi-major axis length, minor axis length, semi-minor axis length, latera recta, length of the latera recta (focal width), focal parameter, eccentricity, linear eccentricity (focal distance), directrices, asymptotes, x-intercepts, y-intercepts, domain, and range of the hyperbola $$$- \left(x - 6\right)^{2} + \frac{\left(y - 5\right)^{2}}{64} = 1$$$.

Solution

The equation of a hyperbola is $$$\frac{\left(y - k\right)^{2}}{b^{2}} - \frac{\left(x - h\right)^{2}}{a^{2}} = 1$$$, where $$$\left(h, k\right)$$$ is the center, $$$a$$$ and $$$b$$$ are the lengths of the semi-major and the semi-minor axes.

Our hyperbola in this form is $$$\frac{\left(y - 5\right)^{2}}{64} - \frac{\left(x - 6\right)^{2}}{1} = 1$$$.

Thus, $$$h = 6$$$, $$$k = 5$$$, $$$a = 1$$$, $$$b = 8$$$.

The standard form is $$$\frac{\left(y - 5\right)^{2}}{8^{2}} - \frac{\left(x - 6\right)^{2}}{1^{2}} = 1$$$.

The vertex form is $$$\frac{\left(y - 5\right)^{2}}{64} - \left(x - 6\right)^{2} = 1$$$.

The general form is $$$64 x^{2} - 768 x - y^{2} + 10 y + 2343 = 0$$$.

The linear eccentricity (focal distance) is $$$c = \sqrt{b^{2} + a^{2}} = \sqrt{65}$$$.

The eccentricity is $$$e = \frac{c}{b} = \frac{\sqrt{65}}{8}$$$.

The first focus is $$$\left(h, k - c\right) = \left(6, 5 - \sqrt{65}\right)$$$.

The second focus is $$$\left(h, k + c\right) = \left(6, 5 + \sqrt{65}\right)$$$.

The first vertex is $$$\left(h, k - b\right) = \left(6, -3\right)$$$.

The second vertex is $$$\left(h, k + b\right) = \left(6, 13\right)$$$.

The first co-vertex is $$$\left(h - a, k\right) = \left(5, 5\right)$$$.

The second co-vertex is $$$\left(h + a, k\right) = \left(7, 5\right)$$$.

The length of the major axis is $$$2 b = 16$$$.

The length of the minor axis is $$$2 a = 2$$$.

The focal parameter is the distance between the focus and the directrix: $$$\frac{a^{2}}{c} = \frac{\sqrt{65}}{65}$$$.

The latera recta are the lines parallel to the minor axis that pass through the foci.

The first latus rectum is $$$y = 5 - \sqrt{65}$$$.

The second latus rectum is $$$y = 5 + \sqrt{65}$$$.

The endpoints of the first latus rectum can be found by solving the system $$$\begin{cases} 64 x^{2} - 768 x - y^{2} + 10 y + 2343 = 0 \\ y = 5 - \sqrt{65} \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the first latus rectum are $$$\left(\frac{47}{8}, 5 - \sqrt{65}\right)$$$, $$$\left(\frac{49}{8}, 5 - \sqrt{65}\right)$$$.

The endpoints of the second latus rectum can be found by solving the system $$$\begin{cases} 64 x^{2} - 768 x - y^{2} + 10 y + 2343 = 0 \\ y = 5 + \sqrt{65} \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the second latus rectum are $$$\left(\frac{47}{8}, 5 + \sqrt{65}\right)$$$, $$$\left(\frac{49}{8}, 5 + \sqrt{65}\right)$$$.

The length of the latera recta (focal width) is $$$\frac{2 a^{2}}{b} = \frac{1}{4}$$$.

The first directrix is $$$y = k - \frac{b^{2}}{c} = 5 - \frac{64 \sqrt{65}}{65}$$$.

The second directrix is $$$y = k + \frac{b^{2}}{c} = \frac{325 + 64 \sqrt{65}}{65}$$$.

The first asymptote is $$$y = - \frac{b}{a} \left(x - h\right) + k = 53 - 8 x$$$.

The second asymptote is $$$y = \frac{b}{a} \left(x - h\right) + k = 8 x - 43$$$.

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).

y-intercepts: $$$\left(0, 5 - 8 \sqrt{37}\right)$$$, $$$\left(0, 5 + 8 \sqrt{37}\right)$$$

Answer

Standard form/equation: $$$\frac{\left(y - 5\right)^{2}}{8^{2}} - \frac{\left(x - 6\right)^{2}}{1^{2}} = 1$$$A.

Vertex form/equation: $$$\frac{\left(y - 5\right)^{2}}{64} - \left(x - 6\right)^{2} = 1$$$A.

General form/equation: $$$64 x^{2} - 768 x - y^{2} + 10 y + 2343 = 0$$$A.

First focus-directrix form/equation: $$$\left(x - 6\right)^{2} + \left(y - 5 + \sqrt{65}\right)^{2} = \frac{65 \left(y - 5 + \frac{64 \sqrt{65}}{65}\right)^{2}}{64}$$$A.

Second focus-directrix form/equation: $$$\left(x - 6\right)^{2} + \left(y - \sqrt{65} - 5\right)^{2} = \frac{65 \left(y - \frac{325 + 64 \sqrt{65}}{65}\right)^{2}}{64}$$$A.

Graph: see the graphing calculator.

Center: $$$\left(6, 5\right)$$$A.

First focus: $$$\left(6, 5 - \sqrt{65}\right)\approx \left(6, -3.06225774829855\right)$$$A.

Second focus: $$$\left(6, 5 + \sqrt{65}\right)\approx \left(6, 13.06225774829855\right)$$$A.

First vertex: $$$\left(6, -3\right)$$$A.

Second vertex: $$$\left(6, 13\right)$$$A.

First co-vertex: $$$\left(5, 5\right)$$$A.

Second co-vertex: $$$\left(7, 5\right)$$$A.

Major (transverse) axis length: $$$16$$$A.

Semi-major axis length: $$$8$$$A.

Minor (conjugate) axis length: $$$2$$$A.

Semi-minor axis length: $$$1$$$A.

First latus rectum: $$$y = 5 - \sqrt{65}\approx -3.06225774829855$$$A.

Second latus rectum: $$$y = 5 + \sqrt{65}\approx 13.06225774829855$$$A.

Endpoints of the first latus rectum: $$$\left(\frac{47}{8}, 5 - \sqrt{65}\right)\approx \left(5.875, -3.06225774829855\right)$$$, $$$\left(\frac{49}{8}, 5 - \sqrt{65}\right)\approx \left(6.125, -3.06225774829855\right)$$$A.

Endpoints of the second latus rectum: $$$\left(\frac{47}{8}, 5 + \sqrt{65}\right)\approx \left(5.875, 13.06225774829855\right)$$$, $$$\left(\frac{49}{8}, 5 + \sqrt{65}\right)\approx \left(6.125, 13.06225774829855\right)$$$A.

Length of the latera recta (focal width): $$$\frac{1}{4} = 0.25$$$A.

Focal parameter: $$$\frac{\sqrt{65}}{65}\approx 0.124034734589208$$$A.

Eccentricity: $$$\frac{\sqrt{65}}{8}\approx 1.007782218537319$$$A.

Linear eccentricity (focal distance): $$$\sqrt{65}\approx 8.06225774829855$$$A.

First directrix: $$$y = 5 - \frac{64 \sqrt{65}}{65}\approx -2.938223013709341$$$A.

Second directrix: $$$y = \frac{325 + 64 \sqrt{65}}{65}\approx 12.938223013709341$$$A.

First asymptote: $$$y = 53 - 8 x$$$A.

Second asymptote: $$$y = 8 x - 43$$$A.

x-intercepts: no x-intercepts.

y-intercepts: $$$\left(0, 5 - 8 \sqrt{37}\right)\approx \left(0, -43.662100242385758\right)$$$, $$$\left(0, 5 + 8 \sqrt{37}\right)\approx \left(0, 53.662100242385758\right)$$$A.

Domain: $$$\left(-\infty, \infty\right)$$$A.

Range: $$$\left(-\infty, -3\right] \cup \left[13, \infty\right)$$$A.

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Properties of the hyperbola $$$- \left(x - 6\right)^{2} + \frac{\left(y - 5\right)^{2}}{64} = 1$$$

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