Properties of the hyperbola $$$16 x^{2} - 4 y^{2} = 64$$$

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 $$$16 x^{2} - 4 y^{2} = 64$$$.

The equation of a hyperbola is $$$\frac{\left(x - h\right)^{2}}{a^{2}} - \frac{\left(y - k\right)^{2}}{b^{2}} = 1$$$, 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 $$$\frac{\left(x - 0\right)^{2}}{4} - \frac{\left(y - 0\right)^{2}}{16} = 1$$$.

Thus, $$$h = 0$$$, $$$k = 0$$$, $$$a = 2$$$, $$$b = 4$$$.

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

The vertex form is $$$\frac{x^{2}}{4} - \frac{y^{2}}{16} = 1$$$.

The general form is $$$4 x^{2} - y^{2} - 16 = 0$$$.

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

The eccentricity is $$$e = \frac{c}{a} = \sqrt{5}$$$.

The first focus is $$$\left(h - c, k\right) = \left(- 2 \sqrt{5}, 0\right)$$$.

The second focus is $$$\left(h + c, k\right) = \left(2 \sqrt{5}, 0\right)$$$.

The first vertex is $$$\left(h - a, k\right) = \left(-2, 0\right)$$$.

The second vertex is $$$\left(h + a, k\right) = \left(2, 0\right)$$$.

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

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

The length of the major axis is $$$2 a = 4$$$.

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

The focal parameter is the distance between the focus and the directrix: $$$\frac{b^{2}}{c} = \frac{8 \sqrt{5}}{5}$$$.

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

The first latus rectum is $$$x = - 2 \sqrt{5}$$$.

The second latus rectum is $$$x = 2 \sqrt{5}$$$.

The endpoints of the first latus rectum can be found by solving the system $$$\begin{cases} 4 x^{2} - y^{2} - 16 = 0 \\ x = - 2 \sqrt{5} \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the first latus rectum are $$$\left(- 2 \sqrt{5}, -8\right)$$$, $$$\left(- 2 \sqrt{5}, 8\right)$$$.

The endpoints of the second latus rectum can be found by solving the system $$$\begin{cases} 4 x^{2} - y^{2} - 16 = 0 \\ x = 2 \sqrt{5} \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the second latus rectum are $$$\left(2 \sqrt{5}, -8\right)$$$, $$$\left(2 \sqrt{5}, 8\right)$$$.

The length of the latera recta (focal width) is $$$\frac{2 b^{2}}{a} = 16$$$.

The first directrix is $$$x = h - \frac{a^{2}}{c} = - \frac{2 \sqrt{5}}{5}$$$.

The second directrix is $$$x = h + \frac{a^{2}}{c} = \frac{2 \sqrt{5}}{5}$$$.

The first asymptote is $$$y = - \frac{b}{a} \left(x - h\right) + k = - 2 x$$$.

The second asymptote is $$$y = \frac{b}{a} \left(x - h\right) + k = 2 x$$$.

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

x-intercepts: $$$\left(-2, 0\right)$$$, $$$\left(2, 0\right)$$$

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.

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

Vertex form/equation: $$$\frac{x^{2}}{4} - \frac{y^{2}}{16} = 1$$$A.

General form/equation: $$$4 x^{2} - y^{2} - 16 = 0$$$A.

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

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

Graph: see the graphing calculator.

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

First focus: $$$\left(- 2 \sqrt{5}, 0\right)\approx \left(-4.472135954999579, 0\right)$$$A.

Second focus: $$$\left(2 \sqrt{5}, 0\right)\approx \left(4.472135954999579, 0\right)$$$A.

First vertex: $$$\left(-2, 0\right)$$$A.

Second vertex: $$$\left(2, 0\right)$$$A.

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

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

Major (transverse) axis length: $$$4$$$A.

Semi-major axis length: $$$2$$$A.

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

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

First latus rectum: $$$x = - 2 \sqrt{5}\approx -4.472135954999579$$$A.

Second latus rectum: $$$x = 2 \sqrt{5}\approx 4.472135954999579$$$A.

Endpoints of the first latus rectum: $$$\left(- 2 \sqrt{5}, -8\right)\approx \left(-4.472135954999579, -8\right)$$$, $$$\left(- 2 \sqrt{5}, 8\right)\approx \left(-4.472135954999579, 8\right)$$$A.

Endpoints of the second latus rectum: $$$\left(2 \sqrt{5}, -8\right)\approx \left(4.472135954999579, -8\right)$$$, $$$\left(2 \sqrt{5}, 8\right)\approx \left(4.472135954999579, 8\right)$$$A.

Length of the latera recta (focal width): $$$16$$$A.

Focal parameter: $$$\frac{8 \sqrt{5}}{5}\approx 3.577708763999664$$$A.

Eccentricity: $$$\sqrt{5}\approx 2.23606797749979$$$A.

Linear eccentricity (focal distance): $$$2 \sqrt{5}\approx 4.472135954999579$$$A.

First directrix: $$$x = - \frac{2 \sqrt{5}}{5}\approx -0.894427190999916$$$A.

Second directrix: $$$x = \frac{2 \sqrt{5}}{5}\approx 0.894427190999916$$$A.

First asymptote: $$$y = - 2 x$$$A.

Second asymptote: $$$y = 2 x$$$A.

x-intercepts: $$$\left(-2, 0\right)$$$, $$$\left(2, 0\right)$$$A.

y-intercepts: no y-intercepts.

Domain: $$$\left(-\infty, -2\right] \cup \left[2, \infty\right)$$$A.

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