Properties of the hyperbola x^2/25 - y^2/49 = 1

The calculator will find the properties of the hyperbola $$$\frac{x^{2}}{25} - \frac{y^{2}}{49} = 1$$$, 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 $$$\frac{x^{2}}{25} - \frac{y^{2}}{49} = 1$$$.

Solution

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 $$$\frac{\left(x - 0\right)^{2}}{25} - \frac{\left(y - 0\right)^{2}}{49} = 1$$$.

Thus, $$$h = 0$$$, $$$k = 0$$$, $$$a = 5$$$, $$$b = 7$$$.

The standard form is $$$\frac{x^{2}}{5^{2}} - \frac{y^{2}}{7^{2}} = 1$$$.

The vertex form is $$$\frac{x^{2}}{25} - \frac{y^{2}}{49} = 1$$$.

The general form is $$$49 x^{2} - 25 y^{2} - 1225 = 0$$$.

The linear eccentricity (focal distance) is $$$c = \sqrt{a^{2} + b^{2}} = \sqrt{74}$$$.

The eccentricity is $$$e = \frac{c}{a} = \frac{\sqrt{74}}{5}$$$.

The first focus is $$$\left(h - c, k\right) = \left(- \sqrt{74}, 0\right)$$$.

The second focus is $$$\left(h + c, k\right) = \left(\sqrt{74}, 0\right)$$$.

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

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

The first co-vertex is $$$\left(h, k - b\right) = \left(0, -7\right)$$$.

The second co-vertex is $$$\left(h, k + b\right) = \left(0, 7\right)$$$.

The length of the major axis is $$$2 a = 10$$$.

The length of the minor axis is $$$2 b = 14$$$.

The focal parameter is the distance between the focus and the directrix: $$$\frac{b^{2}}{c} = \frac{49 \sqrt{74}}{74}$$$.

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

The first latus rectum is $$$x = - \sqrt{74}$$$.

The second latus rectum is $$$x = \sqrt{74}$$$.

The endpoints of the first latus rectum can be found by solving the system $$$\begin{cases} 49 x^{2} - 25 y^{2} - 1225 = 0 \\ x = - \sqrt{74} \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the first latus rectum are $$$\left(- \sqrt{74}, - \frac{49}{5}\right)$$$, $$$\left(- \sqrt{74}, \frac{49}{5}\right)$$$.

The endpoints of the second latus rectum can be found by solving the system $$$\begin{cases} 49 x^{2} - 25 y^{2} - 1225 = 0 \\ x = \sqrt{74} \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the second latus rectum are $$$\left(\sqrt{74}, - \frac{49}{5}\right)$$$, $$$\left(\sqrt{74}, \frac{49}{5}\right)$$$.

The length of the latera recta (focal width) is $$$\frac{2 b^{2}}{a} = \frac{98}{5}$$$.

The first directrix is $$$x = h - \frac{a^{2}}{c} = - \frac{25 \sqrt{74}}{74}$$$.

The second directrix is $$$x = h + \frac{a^{2}}{c} = \frac{25 \sqrt{74}}{74}$$$.

The first asymptote is $$$y = - \frac{b}{a} \left(x - h\right) + k = - \frac{7 x}{5}$$$.

The second asymptote is $$$y = \frac{b}{a} \left(x - h\right) + k = \frac{7 x}{5}$$$.

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

x-intercepts: $$$\left(-5, 0\right)$$$, $$$\left(5, 0\right)$$$

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.

Answer

Standard form/equation: $$$\frac{x^{2}}{5^{2}} - \frac{y^{2}}{7^{2}} = 1$$$A.

Vertex form/equation: $$$\frac{x^{2}}{25} - \frac{y^{2}}{49} = 1$$$A.

General form/equation: $$$49 x^{2} - 25 y^{2} - 1225 = 0$$$A.

First focus-directrix form/equation: $$$\left(x + \sqrt{74}\right)^{2} + y^{2} = \frac{74 \left(x + \frac{25 \sqrt{74}}{74}\right)^{2}}{25}$$$A.

Second focus-directrix form/equation: $$$\left(x - \sqrt{74}\right)^{2} + y^{2} = \frac{74 \left(x - \frac{25 \sqrt{74}}{74}\right)^{2}}{25}$$$A.

Graph: see the graphing calculator.

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

First focus: $$$\left(- \sqrt{74}, 0\right)\approx \left(-8.602325267042627, 0\right)$$$A.

Second focus: $$$\left(\sqrt{74}, 0\right)\approx \left(8.602325267042627, 0\right)$$$A.

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

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

First co-vertex: $$$\left(0, -7\right)$$$A.

Second co-vertex: $$$\left(0, 7\right)$$$A.

Major (transverse) axis length: $$$10$$$A.

Semi-major axis length: $$$5$$$A.

Minor (conjugate) axis length: $$$14$$$A.

Semi-minor axis length: $$$7$$$A.

First latus rectum: $$$x = - \sqrt{74}\approx -8.602325267042627$$$A.

Second latus rectum: $$$x = \sqrt{74}\approx 8.602325267042627$$$A.

Endpoints of the first latus rectum: $$$\left(- \sqrt{74}, - \frac{49}{5}\right)\approx \left(-8.602325267042627, -9.8\right)$$$, $$$\left(- \sqrt{74}, \frac{49}{5}\right)\approx \left(-8.602325267042627, 9.8\right)$$$A.

Endpoints of the second latus rectum: $$$\left(\sqrt{74}, - \frac{49}{5}\right)\approx \left(8.602325267042627, -9.8\right)$$$, $$$\left(\sqrt{74}, \frac{49}{5}\right)\approx \left(8.602325267042627, 9.8\right)$$$A.

Length of the latera recta (focal width): $$$\frac{98}{5} = 19.6$$$A.

Focal parameter: $$$\frac{49 \sqrt{74}}{74}\approx 5.696134298447145$$$A.

Eccentricity: $$$\frac{\sqrt{74}}{5}\approx 1.720465053408525$$$A.

Linear eccentricity (focal distance): $$$\sqrt{74}\approx 8.602325267042627$$$A.

First directrix: $$$x = - \frac{25 \sqrt{74}}{74}\approx -2.906190968595482$$$A.

Second directrix: $$$x = \frac{25 \sqrt{74}}{74}\approx 2.906190968595482$$$A.

First asymptote: $$$y = - \frac{7 x}{5} = - 1.4 x$$$A.

Second asymptote: $$$y = \frac{7 x}{5} = 1.4 x$$$A.

x-intercepts: $$$\left(-5, 0\right)$$$, $$$\left(5, 0\right)$$$A.

y-intercepts: no y-intercepts.

Domain: $$$\left(-\infty, -5\right] \cup \left[5, \infty\right)$$$A.

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

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Properties of the hyperbola $$$\frac{x^{2}}{25} - \frac{y^{2}}{49} = 1$$$

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