Hyperbola
Definition and Properties of a Hyperbola
A hyperbola is a plane curve such that the difference of the distances from any point of the curve to two other fixed points (called the foci of the hyperbola) is constant. The midpoint of the line segment joining the foci is called the center of the hyperbola. The distance from the center to a focus is called the focal distance and denoted by c.
Any hyperbola consists of two distinct branches. The points on the two branches that are closest to each other are called the vertices. The distance of a vertex to the center is denoted by a.
The line passing through the vertices of a hyperbola is called the transverse axis. (It contains the segment of length 2a between the vertices). The transverse axis of a hyperbola is its line of symmetry. Another line of symmetry is perpendicular to the transverse axis and is called the conjugate axis. (It contains the segment of length 2b with midpoint at the center of the hyperbola).
The standard (or canonical) equation of the hyperbola is
The absolute value of the difference of the distances from any point of a hyperbola to its foci is constant:
where
Equations of the Asymptotes of a Hyperbola
Relationship between Semi-Axes of a Hyperbola and its Focal Distance
where
Eccentricity of a Hyperbola
Equations of the Directrices of a Hyperbola
The directrix of a hyperbola is a straight line perpendicular to the transverse axis of the hyperbola and intersecting it at the distance
Parametric Equations of a Hyperbola (Right Branch)
where
General Equation of a Hyperbola
where
General Equation of a Hyperbola with Axes Parallel to the Coordinate Axes
where
Equilateral Hyperbola
A hyperbola is called equilateral if its semi-axes are equal to each other:
If the asymptotes are taken to be the horizontal and vertical coordinate axes (respectively,
Solved Problems
Example 1.
Find the transverse and conjugate semi-axes, foci and eccentricity of the hyperbola
Solution.
Let's write the equation of the hyperbola in canonical form:
Thus the length of the transverse semi-axis is
Calculate the focal distance of the hyperbola
The two foci of the hyperbola have coordinates
Finally the eccentricity is equal
Example 2.
Find the equation of an equilateral hyperbola if it passes through the point
Solution.
The canonical equation of an equilateral hyperbola has the form
Let's substitute the coordinates of the given point and find the value of
Therefore, the equation of the hyperbola is written as
We can also express it like this
Example 3.
Find the equation of a hyperbola if the distance between the foci is 6 and its eccentricity is
Solution.
Let's denote the focal distance of the hyperbola by
where
Let's solve this system and determine the hyperbola parameters
Now we can easily find the conjugate semi-axis
So the canonical equation of the hyperbola has the form
Example 4.
Find the equation of a hyperbola if its asymptotes are the lines
Solution.
The asymptotes of a hyperbola are given by the equations
Hence, we have the first equation
The second equation determines the distance between the foci, that is
where
We also know that
So we have a system of two equations with unknowns
Let's solve this system and determine the parameters of the hyperbola.
Then the equation of the hyperbola is written as
Example 5.
A point with the
Solution.
Let's convert the hyperbola equation into canonical form and define its semi-axes:
Therefore the semi-axes are equal:
Find the
That is, the coordinates of point M are equal
Now let's find the coordinates of the foci
So, the foci have the following coordinates:
Calculate the distances
So the answer is