Conics
Hyperbolas
Learning Objectives
By the end of this section, you will be able to:
- Graph a hyperbola with center at
- Graph a hyperbola with center at
- Identify conic sections by their equations
Before you get started, take this readiness quiz.
Graph a Hyperbola with Center at (0, 0)
The last conic section we will look at is called a hyperbola. We will see that the equation of a hyperbola looks the same as the equation of an ellipse, except it is a difference rather than a sum. While the equations of an ellipse and a hyperbola are very similar, their graphs are very different.
We define a hyperbola as 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.
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.
Again our goal is to connect the geometry of a conic with algebra. Placing the hyperbola on a rectangular coordinate system gives us that opportunity. In the figure, we placed the hyperbola so the foci are on the x-axis and the center is the origin.
The definition states the difference of the distance from the foci to a point is constant. So is a constant that we will call so We will use the distance formula to lead us to an algebraic formula for an ellipse.
To graph the hyperbola, it will be helpful to know about the intercepts. We will find the x-intercepts and y-intercepts using the formula.
The a, b values in the equation also help us find the asymptotes of the hyperbola. The asymptotes are intersecting straight lines that the branches of the graph approach but never intersect as the x, y values get larger and larger.
To find the asymptotes, we sketch a rectangle whose sides intersect the x-axis at the vertices and intersect the y-axis at The lines containing the diagonals of this rectangle are the asymptotes of the hyperbola. The rectangle and asymptotes are not part of the hyperbola, but they help us graph the hyperbola.
The asymptotes pass through the origin and we can evaluate their slope using the rectangle we sketched. They have equations and
There are two equations for hyperbolas, depending whether the transverse axis is vertical or horizontal. We can tell whether the transverse axis is horizontal by looking at the equation. When the equation is in standard form, if the x2-term is positive, the transverse axis is horizontal. When the equation is in standard form, if the y2-term is positive, the transver