Hyperbola
Definition
A hyperbola consists of two curves opening in opposite directions. Just like one of its conic partners, the ellipse, a hyperbola also has two foci and is defined as the set of points where the absolute value of the difference of the distances to the two foci is constant. Thus, every point of a hyperbola whose foci are points and would satisfy where is a constant.
Equation and Analytical Properties
A hyperbola consists of a center, an axis, two vertices, two foci, and two asymptotes. A hyperbola's axis is the line that passes through the two foci, and the center is the midpoint of the two foci. The two vertices are where the hyperbola meets with its axis. On the coordinate plane, we most often use the - or -axis as the hyperbola's axis. The equation for the hyperbola in each of these cases is as follows:
The equation of a hyperbola whose axis is the -axis and whose center is the origin is
Every point of the hyperbola satisfies where the coordinates of the two foci are and and The coordinates of the vertices are and
The equation of a hyperbola whose axis is the -axis and whose center is the origin is
Every point of the hyperbola satisfies where the coordinates of the two foci are and and The coordinates of the vertices are and
By translating these equations, we can express any hyperbola on the coordinate plane whose axis is parallel to either the or -axis.
Now let's discuss the asymptotes of a hyperbola. Every hyperbola has two asymptotes that are symmetrical about the hyperbola's axis. For a hyperbola whose equation is the equations of the asymptotes are
Imagine taking the limit of Then will also increase indefinitely, and the 1 on the right-hand side will be eliminated. Hence the ratio between and will either become or
Example Problems
Find the two foci of the hyperbola
The given hyperbola has the origin as its center and the -axis as its axis. Then the two foci are given by and where Hence, and the two foci are and
What is the equation of a hyperbola whose foci are and and the absolute value of the difference of the distances to the two foci is 4?
Since the two foci are given as and the center of the hyperbola is the origin and its axis is the -axis. Hence we have
What is the equation of a hyperbola whose foci are and and the absolute value of the difference of the distances to the two foci is 6?
Since the two foci are given as and the center of the hyperbola is the origin and its axis is the -axis. Hence we have
What is the equation of a hyperbola whose foci are and and the absolute value of the difference of the distances to the two foci is 4?
Since the axis is the line that passes through the two foci, the axis of the given hyperbola is The center of the hyperbola is the midpoint of the two foci, which is in this case. We can think of the hyperbola as a parallel translation of four units to the right and 3 units upward from Since the absolute value of the difference of the distances to the two foci is 4, it must be true that or Since the distance between the two foci is 6, we know that Hence we have
Therefore the equation of the hyperbola is
The figure below depicts the graph of the hyperbola: where and are its two foci. If the length of is 4, what is the perimeter of
From the equation we have and Since the hyperbola's axis is the -axis, the absolute value of the difference of the distances from the two foci is Since the coordinates of the foci are and We are given so Therefore the perimeter of is
What are the equations of the asymptotes of the hyperbola
For a hyperbola the equation of its asymptotes is given by the formula
Therefore the answer is