Properties of the hyperbola $$$- 9 x^{2} + 16 y^{2} = 144$$$

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 $$$- 9 x^{2} + 16 y^{2} = 144$$$.

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 $$$\frac{\left(y - 0\right)^{2}}{9} - \frac{\left(x - 0\right)^{2}}{16} = 1$$$.

Thus, $$$h = 0$$$, $$$k = 0$$$, $$$a = 4$$$, $$$b = 3$$$.

The standard form is $$$\frac{y^{2}}{3^{2}} - \frac{x^{2}}{4^{2}} = 1$$$.

The vertex form is $$$\frac{y^{2}}{9} - \frac{x^{2}}{16} = 1$$$.

The general form is $$$9 x^{2} - 16 y^{2} + 144 = 0$$$.

The linear eccentricity (focal distance) is $$$c = \sqrt{b^{2} + a^{2}} = 5$$$.

The eccentricity is $$$e = \frac{c}{b} = \frac{5}{3}$$$.

The first focus is $$$\left(h, k - c\right) = \left(0, -5\right)$$$.

The second focus is $$$\left(h, k + c\right) = \left(0, 5\right)$$$.

The first vertex is $$$\left(h, k - b\right) = \left(0, -3\right)$$$.

The second vertex is $$$\left(h, k + b\right) = \left(0, 3\right)$$$.

The first co-vertex is $$$\left(h - a, k\right) = \left(-4, 0\right)$$$.

The second co-vertex is $$$\left(h + a, k\right) = \left(4, 0\right)$$$.

The length of the major axis is $$$2 b = 6$$$.

The length of the minor axis is $$$2 a = 8$$$.

The focal parameter is the distance between the focus and the directrix: $$$\frac{a^{2}}{c} = \frac{16}{5}$$$.

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

The first latus rectum is $$$y = -5$$$.

The second latus rectum is $$$y = 5$$$.

The endpoints of the first latus rectum can be found by solving the system $$$\begin{cases} 9 x^{2} - 16 y^{2} + 144 = 0 \\ y = -5 \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the first latus rectum are $$$\left(- \frac{16}{3}, -5\right)$$$, $$$\left(\frac{16}{3}, -5\right)$$$.

The endpoints of the second latus rectum can be found by solving the system $$$\begin{cases} 9 x^{2} - 16 y^{2} + 144 = 0 \\ y = 5 \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the second latus rectum are $$$\left(- \frac{16}{3}, 5\right)$$$, $$$\left(\frac{16}{3}, 5\right)$$$.

The length of the latera recta (focal width) is $$$\frac{2 a^{2}}{b} = \frac{32}{3}$$$.

The first directrix is $$$y = k - \frac{b^{2}}{c} = - \frac{9}{5}$$$.

The second directrix is $$$y = k + \frac{b^{2}}{c} = \frac{9}{5}$$$.

The first asymptote is $$$y = - \frac{b}{a} \left(x - h\right) + k = - \frac{3 x}{4}$$$.

The second asymptote is $$$y = \frac{b}{a} \left(x - h\right) + k = \frac{3 x}{4}$$$.

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: $$$\left(0, -3\right)$$$, $$$\left(0, 3\right)$$$

Standard form/equation: $$$\frac{y^{2}}{3^{2}} - \frac{x^{2}}{4^{2}} = 1$$$A.

Vertex form/equation: $$$\frac{y^{2}}{9} - \frac{x^{2}}{16} = 1$$$A.

General form/equation: $$$9 x^{2} - 16 y^{2} + 144 = 0$$$A.

First focus-directrix form/equation: $$$x^{2} + \left(y + 5\right)^{2} = \frac{25 \left(y + \frac{9}{5}\right)^{2}}{9}$$$A.

Second focus-directrix form/equation: $$$x^{2} + \left(y - 5\right)^{2} = \frac{25 \left(y - \frac{9}{5}\right)^{2}}{9}$$$A.

Graph: see the graphing calculator.

Center: $$$\left(0, 0\right)$$$A.

First focus: $$$\left(0, -5\right)$$$A.

Second focus: $$$\left(0, 5\right)$$$A.

First vertex: $$$\left(0, -3\right)$$$A.

Second vertex: $$$\left(0, 3\right)$$$A.

First co-vertex: $$$\left(-4, 0\right)$$$A.

Second co-vertex: $$$\left(4, 0\right)$$$A.

Major (transverse) axis length: $$$6$$$A.

Semi-major axis length: $$$3$$$A.

Minor (conjugate) axis length: $$$8$$$A.

Semi-minor axis length: $$$4$$$A.

First latus rectum: $$$y = -5$$$A.

Second latus rectum: $$$y = 5$$$A.

Endpoints of the first latus rectum: $$$\left(- \frac{16}{3}, -5\right)\approx \left(-5.333333333333333, -5\right)$$$, $$$\left(\frac{16}{3}, -5\right)\approx \left(5.333333333333333, -5\right)$$$A.

Endpoints of the second latus rectum: $$$\left(- \frac{16}{3}, 5\right)\approx \left(-5.333333333333333, 5\right)$$$, $$$\left(\frac{16}{3}, 5\right)\approx \left(5.333333333333333, 5\right)$$$A.

Length of the latera recta (focal width): $$$\frac{32}{3}\approx 10.666666666666667$$$A.

Focal parameter: $$$\frac{16}{5} = 3.2$$$A.

Eccentricity: $$$\frac{5}{3}\approx 1.666666666666667$$$A.

Linear eccentricity (focal distance): $$$5$$$A.

First directrix: $$$y = - \frac{9}{5} = -1.8$$$A.

Second directrix: $$$y = \frac{9}{5} = 1.8$$$A.

First asymptote: $$$y = - \frac{3 x}{4} = - 0.75 x$$$A.

Second asymptote: $$$y = \frac{3 x}{4} = 0.75 x$$$A.

x-intercepts: no x-intercepts.

y-intercepts: $$$\left(0, -3\right)$$$, $$$\left(0, 3\right)$$$A.

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

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