Hyperbola Calculator

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 $x^{2} - 4 y^{2} = 36$.

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 $\frac{\left(x - 0\right)^{2}}{36} - \frac{\left(y - 0\right)^{2}}{9} = 1$.

Thus, $h = 0$, $k = 0$, $a = 6$, $b = 3$.

The standard form is $\frac{x^{2}}{6^{2}} - \frac{y^{2}}{3^{2}} = 1$.

The vertex form is $\frac{x^{2}}{36} - \frac{y^{2}}{9} = 1$.

The general form is $x^{2} - 4 y^{2} - 36 = 0$.

The linear eccentricity (focal distance) is $c = \sqrt{a^{2} + b^{2}} = 3 \sqrt{5}$.

The eccentricity is $e = \frac{c}{a} = \frac{\sqrt{5}}{2}$.

The first focus is $\left(h - c, k\right) = \left(- 3 \sqrt{5}, 0\right)$.

The second focus is $\left(h + c, k\right) = \left(3 \sqrt{5}, 0\right)$.

The first vertex is $\left(h - a, k\right) = \left(-6, 0\right)$.

The second vertex is $\left(h + a, k\right) = \left(6, 0\right)$.

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

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

The length of the major axis is $2 a = 12$.

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

The focal parameter is the distance between the focus and the directrix: $\frac{b^{2}}{c} = \frac{3 \sqrt{5}}{5}$.

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

The first latus rectum is $x = - 3 \sqrt{5}$.

The second latus rectum is $x = 3 \sqrt{5}$.

The endpoints of the first latus rectum can be found by solving the system $\begin{cases} x^{2} - 4 y^{2} - 36 = 0 \\ x = - 3 \sqrt{5} \end{cases}$ (for steps, see system of equations calculator).

The endpoints of the first latus rectum are $\left(- 3 \sqrt{5}, - \frac{3}{2}\right)$, $\left(- 3 \sqrt{5}, \frac{3}{2}\right)$.

The endpoints of the second latus rectum can be found by solving the system $\begin{cases} x^{2} - 4 y^{2} - 36 = 0 \\ x = 3 \sqrt{5} \end{cases}$ (for steps, see system of equations calculator).

The endpoints of the second latus rectum are $\left(3 \sqrt{5}, - \frac{3}{2}\right)$, $\left(3 \sqrt{5}, \frac{3}{2}\right)$.

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

The first directrix is $x = h - \frac{a^{2}}{c} = - \frac{12 \sqrt{5}}{5}$.

The second directrix is $x = h + \frac{a^{2}}{c} = \frac{12 \sqrt{5}}{5}$.

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

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

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

x-intercepts: $\left(-6, 0\right)$, $\left(6, 0\right)$

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: $\frac{x^{2}}{6^{2}} - \frac{y^{2}}{3^{2}} = 1$A.

Vertex form/equation: $\frac{x^{2}}{36} - \frac{y^{2}}{9} = 1$A.

General form/equation: $x^{2} - 4 y^{2} - 36 = 0$A.

First focus-directrix form/equation: $\left(x + 3 \sqrt{5}\right)^{2} + y^{2} = \frac{5 \left(x + \frac{12 \sqrt{5}}{5}\right)^{2}}{4}$A.

Second focus-directrix form/equation: $\left(x - 3 \sqrt{5}\right)^{2} + y^{2} = \frac{5 \left(x - \frac{12 \sqrt{5}}{5}\right)^{2}}{4}$A.

Graph: see the graphing calculator.

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

First focus: $\left(- 3 \sqrt{5}, 0\right)\approx \left(-6.708203932499369, 0\right)$A.

Second focus: $\left(3 \sqrt{5}, 0\right)\approx \left(6.708203932499369, 0\right)$A.

First vertex: $\left(-6, 0\right)$A.

Second vertex: $\left(6, 0\right)$A.

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

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

Major (transverse) axis length: $12$A.

Semi-major axis length: $6$A.

Minor (conjugate) axis length: $6$A.

Semi-minor axis length: $3$A.

First latus rectum: $x = - 3 \sqrt{5}\approx -6.708203932499369$A.

Second latus rectum: $x = 3 \sqrt{5}\approx 6.708203932499369$A.

Endpoints of the first latus rectum: $\left(- 3 \sqrt{5}, - \frac{3}{2}\right)\approx \left(-6.708203932499369, -1.5\right)$, $\left(- 3 \sqrt{5}, \frac{3}{2}\right)\approx \left(-6.708203932499369, 1.5\right)$A.

Endpoints of the second latus rectum: $\left(3 \sqrt{5}, - \frac{3}{2}\right)\approx \left(6.708203932499369, -1.5\right)$, $\left(3 \sqrt{5}, \frac{3}{2}\right)\approx \left(6.708203932499369, 1.5\right)$A.

Length of the latera recta (focal width): $3$A.

Focal parameter: $\frac{3 \sqrt{5}}{5}\approx 1.341640786499874$A.

Eccentricity: $\frac{\sqrt{5}}{2}\approx 1.118033988749895$A.

Linear eccentricity (focal distance): $3 \sqrt{5}\approx 6.708203932499369$A.

First directrix: $x = - \frac{12 \sqrt{5}}{5}\approx -5.366563145999495$A.

Second directrix: $x = \frac{12 \sqrt{5}}{5}\approx 5.366563145999495$A.

First asymptote: $y = - \frac{x}{2} = - 0.5 x$A.

Second asymptote: $y = \frac{x}{2} = 0.5 x$A.

x-intercepts: $\left(-6, 0\right)$, $\left(6, 0\right)$A.

y-intercepts: no y-intercepts.

Domain: $\left(-\infty, -6\right] \cup \left[6, \infty\right)$A.

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