Lesson 4: Hyperbolas

Property

A hyperbola is the set of points in the plane, the difference of whose distances from two fixed points (the foci) is a constant. If a cone is cut by a plane parallel to its axis, the intersection is a hyperbola, the only conic section made of two separate pieces, or branches.

Examples

• For a hyperbola with foci at $(-5, 0)$ and $(5, 0)$, the point $(4, 0)$ is on one branch. The difference of its distances from the foci is $|4 - (-5)| - |4 - 5| = 9 - 1 = 8$. Any other point $(x, y)$ on the hyperbola must also satisfy this condition.

• The LORAN navigation system uses hyperbolas. A ship receives signals from two radio towers. The time delay between signals places the ship on a specific hyperbola. A third tower provides a second hyperbola, and the ship's location is their intersection.

Property

A hyperbola is all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a focus of the hyperbola.
The line through the foci is called the transverse axis. The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola. The midpoint of the segment joining the foci is called the center of the hyperbola. The line perpendicular to the transverse axis that passes through the center is called the conjugate axis. Each piece of the graph is called a branch of the hyperbola.

Examples

• A point on a hyperbola is 12 units from one focus and 5 units from the other. The constant difference for this hyperbola is $|12 - 5| = 7$.
• The line segment connecting a hyperbola's two vertices is part of the transverse axis, and its midpoint is the center of the hyperbola.
• A hyperbola consists of two separate, mirror-image curves called branches. These branches open infinitely, approaching but never touching the asymptotes.

Explanation

A hyperbola has two U-shaped curves called branches that open away from each other. It's defined by the constant difference in distances from any point on the curve to two fixed points (foci), unlike an ellipse, which uses a constant sum.

Property

The standard form of the equation of a hyperbola with center $(0, 0)$ is

For $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the transverse axis is on the x-axis, it opens left and right, vertices are $(\pm a, 0)$, and asymptotes are $y = \pm \frac{b}{a}x$. For $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, the transverse axis is on the y-axis, it opens up and down, vertices are $(0, \pm a)$, and asymptotes are $y = \pm \frac{a}{b}x$.
To graph: Write the equation in standard form. Find vertices from the positive term's denominator, $a^2$. Sketch a rectangle using points $(\pm a, 0)$ and $(0, \pm b)$ (or vice versa depending on orientation). Draw asymptotes through the rectangle's diagonals. Draw the branches from the vertices, approaching the asymptotes.

Examples

• To graph $\frac{x^2}{16} - \frac{y^2}{9} = 1$, note the center is $(0,0)$ and it opens horizontally since $x^2$ is positive. The vertices are $(\pm 4, 0)$. The guide rectangle uses $(0, \pm 3)$, giving asymptotes $y = \pm \frac{3}{4}x$.
• To graph $\frac{y^2}{64} - \frac{x^2}{25} = 1$, note the center is $(0,0)$ and it opens vertically since $y^2$ is positive. The vertices are $(0, \pm 8)$. The guide rectangle uses $(\pm 5, 0)$, giving asymptotes $y = \pm \frac{8}{5}x$.
• To graph $4y^2 - 25x^2 = 100$, first divide by 100 to get $\frac{y^2}{25} - \frac{x^2}{4} = 1$. This is a vertical hyperbola with vertices at $(0, \pm 5)$ and asymptotes $y = \pm \frac{5}{2}x$.

Explanation

This equation describes a hyperbola centered at the origin. The term with the positive coefficient ($x^2$ or $y^2$) tells you if it opens horizontally or vertically. The values $a$ and $b$ create a guide rectangle for sketching the asymptotes.