Definition, vocabulary and graphic approach
The distance between two points \left(x_0,y_0\right) and \left(x_1,y_1\right) is provided by the distance formula:
\sqrt{\left(x_1-x_0\right)^2+\left(y_1-y_0\right)^2}
The distance between the point \left(2{,}3\right) and the point \left(-2{,}4\right) is:
\sqrt{\left(-2-2\right)^2+\left(4-3\right)^2}=\sqrt{17}
Hyperbolas are often realized as the graphs of inverse functions such as f\left(x\right)=\dfrac{1}{x}. However, not all hyperbolas are graphs of functions.
Hyperbola
A hyperbola is defined in terms of two foci points A,B and a constant c. A point is on the hyperbola if and only if the difference between the distances from the point on the hyperbola to A and B is c.
Properties of hyperbolas
Center, vertices, and foci of a hyperbola
Center of a hyperbola
The center C of a hyperbola is the midpoint of the foci of the hyperbola.
Vertices of a hyperbola
The line segment joining the foci of a hyperbola intersect the hyperbola in two points. These two points, V and W, are the vertices of the hyperbola.
Transverse axis
The transverse axis of a hyperbola is the line segment joining the two vertices of the hyperbola.
The midpoint of the transverse axis of a hyperbola is the center of the hyperbola.
Conjugate axis
The line passing through the center of a hyperbola and perpendicular to the transverse axis is the conjugate axis of the hyperbola.
A hyperbola is unchanged when reflected across its conjugate axis.
Lengths of the transverse axis
Length of transverse axis
Suppose the distance from the center of a hyperbola to a vertex of the hyperbola is a. Then the length of the transverse axis is 2a.
Consider the hyperbola in the following graphic:
The distance from the center of the hyperbola to one of its vertices is:
\sqrt{\left(2-1\right)^2+\left(3-0\right)^2}=\sqrt{10}
Therefore, the length of the transverse axis is:
2\sqrt{10}
Eccentricity
Eccentricity
Consider a hyperbola. There is a number e called the eccentricity of the hyperbola such that if P is any point on the hyperbola, a is the distance from the point P to the nearest focus of the hyperbola and b is the distance from the point P to the conjugate axis of the hyperbola, then:
\dfrac{a}{b}=e
The following graphic contains three hyperbolas with different values for eccentricity.
Smaller values of eccentricity correspond to more "curviness" of the hyperbola around its foci.
Equation of hyperbolas
For the remainder of this section, we will focus on the equations whose graphs are hyperbolas with conjugate axis as either a horizontal or vertical line.
The graph of the function f\left(x\right)=\dfrac{1}{x} is a hyperbola whose conjugate axis is the line y=-x.
Equation of a hyperbola I
Consider the following equation:
\dfrac{\left(x-h\right)^2}{a^2}-\dfrac{\left(y-k\right)^2}{b^2}=1
The set of points in the xy -plane which satisfy the above equation is a hyperbola with the following properties:
- The center of the hyperbola is \left(h,k\right).
- The foci of the hyperbola are at \left(h+2a,k\right) and \left(h-2a,k\right).
- The vertices of the hyperbola are at \left(h+a,k\right) and \left(h-a,k\right).
The above equation can only be used to describe hyperbolas whose conjugate axis is a vertical line. For example, the hyperbola defined by f\left(x\right)=\dfrac{1}{x} is not covered by this theorem.
Consider the following equation:
\dfrac{\left(x-1\right)^2}{4}-\dfrac{\left(y-2\right)^2}{9}=1
The set of points in the xy -plane is a hyperbola with the following properties:
- The center of the hyperbola is \left(1{,}2\right).
- The foci of the hyperbola are at \left(5{,}2\right) and \left(-3{,}2\right).
- The vertices of the hyperbola are at \left(3{,}2\right) and \left(-1{,}2\right).
Equation of a hyperbola II
Consider the following equation:
\dfrac{\left(y-h\right)^2}{a^2}-\dfrac{\left(x-k\right)^2}{b^2}=1
The set of points in the xy -plane which satisfy the above equation is a hyperbola with the following properties:
- The center of the hyperbola is \left(k,h\right).
- The foci of the hyperbola are at \left(k,h+2a\right) and \left(k,h-2a\right).
- The vertices of the hyperbola are at \left(k,h+a\right) and \left(k,h-a\right).
The above equation can only be used to describe hyperbolas whose conjugate axis is a horizontal line.
Consider the following equation:
\dfrac{\left(y-3\right)^2}{9}-\dfrac{\left(x-3\right)^2}{9}=1
Then the set of points in the xy -plane is a hyperbola with the following properties:
- The center of the hyperbola is \left(3{,}3\right)
- The foci of the hyperbola are at \left(3{,}7\right) and \left(3,-1\right)
- The vertices of the hyperbola are at \left(3{,}5\right) and \left(3{,}1\right)
